CN105159077A - Finite-time continuous sliding mode control method for disturbance compensation of direct drive motor system - Google Patents

Abstract

The invention discloses a finite-time continuous sliding mode control method for the disturbance compensation of a direct drive motor system. The method comprises the steps of establishing a mathematical model for the direct drive motor system; designing a disturbance observer for the direct drive motor system and proving the observation accuracy of the disturbance observer; and designing a disturbance observer-based finite-time convergence type continuous sliding mode controller. According to the technical scheme of the invention, the buffeting phenomenon of the sliding mode control is avoided. Meanwhile, the steady-state performance of the system with no finite-time tracking error even under the effect of the interference is still enabled. Moreover, the capability of the sliding mode control method in standing up to the interference of the direct drive motor system and eliminating the buffeting phenomenon of the sliding mode control is enhanced. In addition, the good tracking performance is realized.

Description

The continuous sliding-mode control of finite time that direct driving motor system interference compensates

Technical field

The present invention relates to electromechanical servo control technology field, relate generally to a kind of continuous sliding-mode control of finite time of interference compensation of direct driving motor system.

Background technology

In modern industry is produced, direct driving motor system widely uses as backlash, the by force nonlinear problem such as inertial load and structural flexibility owing to eliminating some mechanical drive problems relevant to reduction gearing in many plant equipment.These nonlinear problems are all the principal elements of influential system performance, and its existence will severe exacerbation system keeps track performance, therefore can obtain high-precision control performance by carrying out advanced Controller gain variations to direct driving motor system.But, also just because of the effect lacking reduction gearing, need when carrying out Controller gain variations to direct driving motor system to face many outer interference, as Parameter Perturbation and outer load disturbance etc., these outer interference no longer directly act on driver part through reduction gearing, seriously control performance can be worsened equally like this, even system order reduction, unstability can be made.Therefore advanced controller design method is explored to ensure that the high precision control performance of direct driving motor system is still the active demand in practical engineering application field.

There is the problem of outer interference for direct driving motor system, many methods are suggested in succession.Wherein sliding-mode control is a kind of very effective method for the problem of the outer interference of process.The basic ideas of sliding-mode control are the nominal plant model CONTROLLER DESIGN for direct driving motor system, the deviation between real system model and nominal plant model are unified to be referred in outer interference with disturbing.For outer interference, traditional sliding-mode control mainly overcomes outer interference by the robustness increasing controller thus urgent system state arrives sliding-mode surface, but, the method increasing the robustness of controller by the method increasing discontinuous term gain increases the buffeting that sliding formwork controls, in practice, probably activating system high frequency is dynamic, makes system unstability.Thus traditional sliding-mode control has very large engineering limitations.Meanwhile, traditional sliding-mode control can only obtain the steady-state behaviour of asymptotic tracking.And in engineering reality, can not be tending towards infinitely great the action time of controller, thus tracking error can not go to zero.

Summary of the invention

The object of the present invention is to provide the continuous sliding-mode control of finite time that a kind of direct driving motor system interference compensates.

The technical solution realizing the object of the invention is: a kind of continuous sliding-mode control of finite time of interference compensation of direct driving motor system, comprises the following steps:

Step 1, sets up the mathematical model of direct driving motor system;

Step 2, the interference observer of design direct driving motor system;

Step 3, designs the continuous sliding mode controller of the finite time convergence control based on interference observer.

Compared with prior art, its remarkable advantage is in the present invention:

(1) observer sliding-mode surface and controller sliding-mode surface combine by the present invention, eliminate the observational error of interference, ensure that the transient state control performance of controller;

(2) the present invention compensate for the outer interference of direct driving motor system, devises continuous sliding mode controller simultaneously, makes the serialization of controller curve, eliminates the buffeting problem that sliding formwork controls, ensure that the robustness of sliding mode control strategy simultaneously;

(3) the present invention does not require that the mathematic(al) representation that system is disturbed exists derivative outward, still can ensure good control performance for the non-existent outer interference of the derivative that may exist;

(4) the present invention finally can obtain the steady-state behaviour that tracking error finite time is zero, ensure that tracking error is zero in finite time.

Accompanying drawing explanation

Fig. 1 is the continuous sliding-mode control process flow diagram of finite time of the interference compensation of direct driving motor system of the present invention.

Fig. 2 is the schematic diagram of direct driving motor system of the present invention.

Fig. 3 is the continuous sliding-mode control of finite time (UCFT ?the SMC) principle schematic of the interference compensation of direct driving motor system.

Fig. 4 be in the embodiment of the present invention UCFT ?under SMC controller action system export the trace plot to expecting instruction.

Fig. 5 be in the embodiment of the present invention UCFT ?the tracking error curve map over time of system under SMC controller action.

Fig. 6 be in the embodiment of the present invention sliding formwork interference observer to the observation curve figure of system interference.

Fig. 7 be in the embodiment of the present invention sliding formwork interference observer to the observational error of system interference curve map over time.

Fig. 8 be in the embodiment of the present invention UFTC ?direct driving motor Systematical control input time history plot under SMC controller action.

Fig. 9 is that in the embodiment of the present invention, under SMC controller action, direct driving motor Systematical control inputs time history plot.

Figure 10 be in the embodiment of the present invention UCFT ?SMC, SMC controller act on the correlation curve figure of lower system tracking error respectively.

Embodiment

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

Composition graphs 1, the continuous sliding-mode control of finite time that direct driving motor system interference of the present invention compensates, comprises the following steps:

Step 1, sets up the mathematical model of direct driving motor system;

Step 1 ?1, the direct driving motor system considered of the present invention is permanent magnet DC motor Direct driver inertia load by being furnished with electrical driver.Composition graphs 2, servomotor output terminal drives inertia load, power supply is powered to servomotor by electrical driver, steering order is by the motion of electrical equipment driver control servomotor, photoelectric encoder feeds back motor position signal to controller, consider that electromagnetic time constant is more much smaller than mechanical time constant, and electric current loop speed is much larger than the response speed of speed ring and position ring, therefore electric current loop can be approximately proportional component;

Therefore, according to Newton second law, the equation of motion of direct driving motor system is:

$m\stackrel{\·\·}{y}=k_{i}u-B\stackrel{\·}{y}+f(t,y,\stackrel{\·}{y})---\left(1\right)$

In formula (1), m is inertia load parameter, k _ifor torque error constant, B is viscosity friction coefficient, be modeling error, comprise m, k _i, 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, u is the control inputs of system, and t is time variable;

Step 1 ?2, definition status variable: then formula (1) equation of motion is converted into state equation:

$\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2={\mathrm{\θ}}_{1}u-{\mathrm{\θ}}_2{x}_2+d(x,t)\\ y=x_1\end{array}---\left(2\right)$

In formula (2), be nominal value and known. can think that system is always disturbed, comprise outer load disturbance, non-modeling friction, Unmarried pregnancy, system actual parameter and modeling parameters depart from the uncertainty caused.F (t, x ₁, x ₂) be above-mentioned x ₁represent the displacement of inertia load, x ₂represent the speed of inertia load;

Because in direct driving motor system, the state of system and parameter are all bounded, therefore the total interference volume d (x, t) of system meets:

|d(x,t)|≤D(3)

In formula (3), D is known normal number, and namely d (x, t) has the known upper bound.

Step 2, design interference observer also proves the accuracy of observation:

Step 2 ?1, design interference observer:

Definition observer sliding-mode surface s ₁for:

s ₁＝z ₁-x ₂(4)

Wherein, z ₁for dynamic in observer;

${\stackrel{\·}{z}}_1=-k_1{s}_1-{\mathrm{\β}}_{1}sign\left(s_1\right)-{\mathrm{\ϵ}}_1{s}_1^{p_1/q_1}-\left|{\mathrm{\θ}}_2{x}_2\right|sign\left(s_1\right)+{\mathrm{\θ}}_{1}u---\left(5\right)$

In formula (5), k ₁, β ₁, ε ₁, p ₁and q ₁be interference observer coefficient; p ₁<q ₁, and be positive odd number, k ₁, β ₁, ε ₁be positive number, β ₁>=D;

sign(0)∈[-1,1]

The then estimation of d (x, t) for:

$\hat{d}(x,t)=-k_1{s}_1-{\mathrm{\&beta;}}_{1}sign\left(s_1\right)-{\mathrm{\&epsiv;}}_1{s}_1^{p_1/q_1}-\left|{\mathrm{\&theta;}}_2{x}_2\right|sign\left(s_1\right)+{\mathrm{\&theta;}}_2{x}_2---\left(7\right)$

Had by formula (2), (4), (5):

$\begin{array}{c}{\stackrel{\&CenterDot;}{s}}_1={\stackrel{\&CenterDot;}{z}}_1-{\stackrel{\&CenterDot;}{x}}_2\\ =-k_1{s}_1-{\mathrm{\&beta;}}_{1}sign\left(s_1\right)-{\mathrm{\&epsiv;}}_1{s}_1^{p_1/q_1}-\left|{\mathrm{\&theta;}}_2{x}_2\right|sign\left(s_1\right)+{\mathrm{\&theta;}}_2{x}_2-d(x,t)\end{array}---\left(8\right)$

Step 2 ?2, definition interference observer Lyapunov Equation:

$V_1\left(t\right)=\frac{1}{2}{s}_1^2---\left(9\right)$

Again because of β ₁>=D, then:

$\begin{array}{c}{\stackrel{\&CenterDot;}{V}}_1\left(t\right)=s_1{\stackrel{\&CenterDot;}{s}}_1\\ =s_1\&lsqb;-k_1{s}_1-{\mathrm{\&beta;}}_{1}sign\left(s_1\right)-{\mathrm{\&epsiv;}}_1{s}_1^{p_1/q_1}-\left|{\mathrm{\&theta;}}_2{x}_2\right|sign\left(s_1\right)+{\mathrm{\&theta;}}_2{x}_2-d(x,t)\&rsqb;\\ =-k_1{s}_1^2-{\mathrm{\&beta;}}_1{s}_{1}sign\left(s_1\right)-{\mathrm{\&epsiv;}}_1{s}_1^{(p_1/q_1)q_1}-\left|{\mathrm{\&theta;}}_2{x}_2\right|\left|s_1\right|+{\mathrm{\&theta;}}_2{x}_2{s}_2-d(x,t)s_1\\ \&le;-k_1{s}_1^2-{\mathrm{\&beta;}}_1\left|s_1\right|-{\mathrm{\&epsiv;}}_1{s}_1^{(p_1/q_1)q_1}+d(x,t)s_1\\ \&le;-k_1{s}_1^2-{\mathrm{\&epsiv;}}_1{s}_1^{(p_1/q_1)q_1}=-2k_1{V}_1\left(t\right)-2^{(p_1/q_1)/2q_1}{\mathrm{\&epsiv;}}_1{V}_1^{(p_1/q_1)/2q_1}\left(t\right)\end{array}---\left(10\right)$

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

${\stackrel{\&CenterDot;}{V}}_0\left(t\right)+{\mathrm{\&alpha;V}}_0\left(t\right)+{\mathrm{\&lambda;V}}_0^{\mathrm{\&gamma;}}\left(t\right)\&le;0,\&ForAll;t>t_0---\left(11\right)$

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

$t_s\&le;t_0+\frac{1}{\mathrm{\&alpha;}(1+\mathrm{\&gamma;})}ln\frac{{\mathrm{\&alpha;V}}_0^{1-\mathrm{\&gamma;}}\left(t_0\right)+\mathrm{\&lambda;}}{\mathrm{\&lambda;}}---\left(12\right)$

Wherein, α >0, λ >0,0< γ <1;

Therefore, V ₁at Finite-time convergence to equilibrium point, namely will there is a time t in (t) ₂point, at t ₂afterwards, V ₁t () perseverance is zero, by V ₁t the expression formula (9) of () is known, V ₁t () is after zero, s ₁also be zero, now also zero will be converged to, again because of d (x, t) evaluated error

$\begin{array}{c}\tilde{d}(x,t)=\hat{d}(x,t)-d(x,t)\\ =-k_1{s}_1-{\mathrm{\&beta;}}_{1}sign\left(s_1\right)-{\mathrm{\&epsiv;}}_1{s}_1^{p_1/q_1}-\left|{\mathrm{\&theta;}}_2{x}_2\right|sign\left(s_1\right)+{\mathrm{\&theta;}}_2{x}_2-d(x,t)\\ =-k_1{s}_1-{\mathrm{\&beta;}}_{1}sign\left(s_1\right)-{\mathrm{\&epsiv;}}_1{s}_1^{p_1/q_1}-\left|{\mathrm{\&theta;}}_2{x}_2\right|sign\left(s_1\right)+{\mathrm{\&theta;}}_2{x}_2-{\stackrel{\&CenterDot;}{x}}_2+{\mathrm{\&theta;}}_{1}u-{\mathrm{\&theta;}}_2{x}_2\\ =-k_1{s}_1-{\mathrm{\&beta;}}_{1}sign\left(s_1\right)-{\mathrm{\&epsiv;}}_1{s}_1^{p_1/q_1}-\left|{\mathrm{\&theta;}}_2{x}_2\right|sign\left(s_1\right)-{\stackrel{\&CenterDot;}{x}}_2+{\mathrm{\&theta;}}_{1}u\\ ={\stackrel{\&CenterDot;}{z}}_1-{\stackrel{\&CenterDot;}{x}}_2\\ ={\stackrel{\&CenterDot;}{s}}_1\end{array}---\left(13\right)$

The evaluated error then disturbed also will at finite time t ₂be inside 0; Namely at t ₂after have

Obtain interference observer:

$\hat{d}(x,t)=-k_1{s}_1-{\mathrm{\&beta;}}_{1}sign\left(s_1\right)-{\mathrm{\&epsiv;}}_1{s}_1^{p_1/q_1}-\left|{\mathrm{\&theta;}}_2{x}_2\right|sign\left(s_1\right)+{\mathrm{\&theta;}}_2{x}_2.$

Step 3, designs the continuous sliding mode controller of the finite time convergence control based on interference observer:

Definition direct driving motor alliance tracking error e ₀(t), error variance e ₁(t):

e ₀(t)＝x ₁-x _d(t)(14)

$e_1\left(t\right)=x_2-{\stackrel{\&CenterDot;}{x}}_d\left(t\right)+s_1---\left(15\right)$

Wherein, x _dt () is system reference position signalling, x _dt () is Second Order Continuous, and system reference position signalling x _d(t), system reference rate signal system reference acceleration signal all bounded;

Definition sliding mode controller sliding-mode surface s:

$s=e_1\left(t\right)+\&Integral;{\mathrm{\&lambda;}}_{0}sign\left(e_0\right(t\left)\right)|e_0\left(t\right){|}^{{\mathrm{\&alpha;}}_1}+{\mathrm{\&lambda;}}_{1}sign\left(e_1\right(t\left)\right)|e_1\left(t\right){|}^{{\mathrm{\&alpha;}}_2}dt---\left(16\right)$

Wherein λ ₀, λ ₁, α ₁, α ₂be sliding mode controller parameter, and be all greater than zero, and λ ₀, λ ₁meet expression formula z ²+ λ ₁z+ λ ₀hurwitz, wherein _zfor differentiating operator, α ₁, α ₂meet α ₂∈ (0,1); Then,

$\begin{array}{c}\stackrel{\&CenterDot;}{s}=s={\stackrel{\&CenterDot;}{e}}_1\left(t\right)+{\mathrm{\&lambda;}}_{0}sign\left(e_0\right(t\left)\right)|e_0\left(t\right){|}^{{\mathrm{\&alpha;}}_1}+{\mathrm{\&lambda;}}_{1}sign\left(e_1\right(t\left)\right)|e_1\left(t\right){|}^{{\mathrm{\&alpha;}}_2}\\ ={\mathrm{\&theta;}}_{1}u-{\mathrm{\&theta;}}_2{x}_2+d(x,t)-{\stackrel{\&CenterDot;\&CenterDot;}{x}}_d+{\mathrm{\&lambda;}}_{0}sign\left(e_0\right(t\left)\right)|e_0\left(t\right){|}^{{\mathrm{\&alpha;}}_1}\\ +{\mathrm{\&lambda;}}_{1}sign\left(e_1\right(t\left)\right)|e_1\left(t\right){|}^{{\mathrm{\&alpha;}}_2}+{\stackrel{\&CenterDot;}{s}}_1\end{array}---\left(17\right)$

Obtaining sliding mode controller u is:

$\begin{array}{c}u=-\frac{1}{{\mathrm{\&theta;}}_1}\&lsqb;-{\mathrm{\&theta;}}_2{x}_2+d(x,t)-{\stackrel{\&CenterDot;\&CenterDot;}{x}}_d+{\mathrm{\&lambda;}}_{0}sign\left(e_0\right(t\left)\right)|e_0\left(t\right){|}^{{\mathrm{\&alpha;}}_1}+{\mathrm{\&lambda;}}_{1}sign\left(e_1\right(t\left)\right)|e_1\left(t\right){|}^{{\mathrm{\&alpha;}}_2}\\ +{\mathrm{\&lambda;}}_{2}s+{\mathrm{\&lambda;}}_{3}sign\left(s\right)|s{|}^{{\mathrm{\&alpha;}}_0}\&rsqb;\end{array}---\left(18\right)$

Wherein λ ₂, λ ₃, α ₀for controller parameter, and λ ₂>0, λ ₃>0,0< α ₀<1.

Step 4 is zero test in system Existence of Global Stable and error finite time:

Formula (18) is substituted into formula (17) have:

$\begin{array}{c}\stackrel{\&CenterDot;}{s}=d\left(t\right)-\hat{d}\left(t\right)+{\stackrel{\&CenterDot;}{s}}_1-{\mathrm{\&lambda;}}_{2}s+{\mathrm{\&lambda;}}_{3}sign\left(s\right)|s{|}^{{\mathrm{\&alpha;}}_0}\\ =-{\mathrm{\&lambda;}}_{2}s-{\mathrm{\&lambda;}}_{3}sign\left(s\right)|s{|}^{{\mathrm{\&alpha;}}_0}\end{array}---\left(19\right)$

Definition sliding mode controller Lyapunov Equation:

$V\left(t\right)=\frac{1}{2}{s}^2---\left(20\right)$

Then have:

$\begin{array}{c}\stackrel{\&CenterDot;}{V}\left(t\right)=s\stackrel{\&CenterDot;}{s}\\ =s(-{\mathrm{\&lambda;}}_{2}s-{\mathrm{\&lambda;}}_{3}sign(s)\left|s{|}^{{\mathrm{\&alpha;}}_0}\right)\\ =-2{\mathrm{\&lambda;}}_{2}V\left(t\right)-2^{({\mathrm{\&alpha;}}_0+1)/2}{\mathrm{\&lambda;}}_3{V}^{({\mathrm{\&alpha;}}_0+1)/2}\left(t\right)\end{array}---\left(21\right)$

Then sliding mode controller sliding-mode surface s will be zero in finite time, namely there is a time point t ₁, at t ₁there is s=0 afterwards, from formula (16), now have:

${\stackrel{\&CenterDot;}{e}}_1\left(t\right)={\mathrm{\&lambda;}}_{0}sign\left(e_0\right(t\left)\right)|e_0\left(t\right){|}^{{\mathrm{\&alpha;}}_1}+{\mathrm{\&lambda;}}_{1}sign\left(e_1\right(t\left)\right)|e_1\left(t\right){|}^{{\mathrm{\&alpha;}}_2}---\left(22\right)$

Then have:

$\begin{array}{c}{\stackrel{\&CenterDot;}{e}}_0\left(t\right)=e_1\left(t\right)-s_1\\ {\stackrel{\&CenterDot;}{e}}_1\left(t\right)={\mathrm{\&lambda;}}_{0}sign\left(e_0\right(t\left)\right)|e_0\left(t\right){|}^{{\mathrm{\&alpha;}}_1}+{\mathrm{\&lambda;}}_{1}sign\left(e_1\right(t\left)\right)|e_1\left(t\right){|}^{{\mathrm{\&alpha;}}_2}\end{array}---\left(23\right)$

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, then there is t ₃=max{t ₁, t ₂, through t ₃have after moment:

$\begin{array}{c}{\stackrel{\&CenterDot;}{e}}_0\left(t\right)=e_1\left(t\right)\\ {\stackrel{\&CenterDot;}{e}}_1\left(t\right)={\mathrm{\&lambda;}}_{0}sign\left(e_0\right(t\left)\right)|e_0\left(t\right){|}^{{\mathrm{\&alpha;}}_1}+{\mathrm{\&lambda;}}_{1}sign\left(e_1\right(t\left)\right)|e_1\left(t\right){|}^{{\mathrm{\&alpha;}}_2}\end{array}---\left(24\right)$

If again because there is a system such as formula shown in (24), and each parameter lambda ₀, λ ₁, α ₁, α ₂all be greater than zero, and λ ₀, λ ₁meet expression formula z ²+ λ ₁z+ λ ₀hurwitz, wherein _zfor differentiating operator, α ₁, α ₂meet α ₂∈ (0,1), then this system state e ₀(t), e ₁t () will be stabilized to equilibrium point in finite time, i.e. once-existing some t ₄, e ₀(t), e ₁t () will at t ₄converging to zero afterwards, is i.e. zero in the tracking error finite time of system.

In summary, it is the result of zero that the continuous sliding-mode control of finite time of the interference compensation designed for direct driving motor system (2) makes system obtain tracking error in finite time, regulates observer coefficient k ₁, β ₁, ε ₁, p ₁, q ₁, the tracking error of observer can be made to go to zero in finite time, adjustment control parameter lambda ₀, λ ₁, α ₁, α ₂, λ ₂, λ ₃, α ₀the tracking error of system can be made to go to zero in finite time.The continuous sliding mode controller principle schematic of finite time that direct driving motor system interference compensates as shown in Figure 3.By the system state x obtained ₁, x ₂, expect trace command x _dorecontrolling factor device sliding-mode surface s and viewer sliding-mode surface s ₁by the outer interference of sliding formwork interference observer observation direct driving motor system, the outer interference observed is passed to the continuous sliding mode controller of finite time of interference compensation, controller is applied in motor driver after calculating controlled quentity controlled variable, thus controls direct driving motor tracking expectation instruction.

Embodiment

For examining designed controller performance, getting following parameter in simulations and modeling carried out to direct driving motor system:

Inertia load parameter m=0.00026kgm ²; Viscosity friction coefficient B=0.00143ms/rad; Torque error constant k _u=1.11Nm/V;

To the expectation instruction of fixed system be: x _d=20sin (t) [1-exp (-0.01t ³)] o;

Interference level: d (x, t)=(0.1/0.00026) sin (0.5 π t) [1-exp (-0.01t ³)] Nm.

Get following controller to compare:

The continuous sliding formwork of the finite time stability of interference compensation controls (UCFT ?SMC) controller: get interference observer parameter k ₁=5000, β ₁=500, ε ₁=0.05, p ₁=3 and q ₁=5; Controller parameter λ ₀=32, λ ₁=36, α ₁=0.25, α ₂=0.4, λ ₂=56, λ ₃=60, α ₀=0.3.

Sliding mode controller (SMC): in order to force system state to arrive sliding-mode surface, the controller parameter chosen is λ ₀=32, λ ₁=36, k=40.

Model uncertainty d (x, t)=(0.1/0.00026) sin (0.5 π t) [1-exp (-0.01t is there is in system ³)] Nm time, UC ?under SMC controller action system export the tracking expecting instruction, tracking error curve as shown in Fig. 4, Fig. 5; Under expecting instruction and UCFT ?SMC controller action in Fig. 4, system curve of output almost overlaps, meanwhile, composition graphs 5, known, under UCFT ?SMC controller action, system has good tracking performance, and steady track error is in 0.08 °;

Fig. 6, Fig. 7 be UCFT ?disturbance-observer curve and observational error change curve in time under SMC controller action, in Fig. 6, disturbance-observer curve overlaps substantially with actual interference curve in system; As can be seen from Fig. 6, Fig. 7, the uncertain observation of model that designed interference observer exists system is very accurate, and as shown in Figure 7, converges in less boundary rapidly, be about 0.15Nm in observational error after the very short time.

Fig. 8, Fig. 9 be UCFT ?under SMC controller action and the control inputs curve over time of the lower system of traditional sliding mode controller (SMC) effect.Traditional sliding-mode control mainly overcomes model uncertainty by the robustness increasing controller thus urgent system state arrives sliding-mode surface, because discontinuous term gain value is larger, as can be seen from Figure 9, significantly buffeting has appearred in the control inputs of system, this may excite high frequency dynamic in Practical Project uses, and system will be made time serious to disperse.And the interference that the continuous sliding-mode control of the finite time with interference compensation (UCFT ?SMC) exists owing to compensate for system, make control inputs serialization simultaneously, thus eliminate the discontinuous term of sliding mode controller theoretically, eliminate the buffeting that sliding formwork controls.Also can find out from Fig. 8, the control inputs of system is a low frequency and continuous curve, do not exist high frequency buffet, be convenient to use in engineering reality, thus demonstrate UCFT ?SMC can eliminate sliding formwork control buffeting problem.

Figure 10 be UCFT ?under SMC controller action and the tracking error curve over time of the lower system of traditional sliding mode controller (SMC) effect.From Figure 10 the tracking error contrast of two kinds of controllers can find out UCFT proposed by the invention ?the tracking error of SMC controller little compared to SMC controller, UCFT ?the amplitude of tracking error of SMC controller be about 0.08 °, the amplitude of the steady track error of SMC controller is about 0.1 °.

Claims (6)

1. the continuous sliding-mode control of finite time of direct driving motor system interference compensation, is characterized in that, comprise the following steps:

Step 1, sets up the mathematical model of direct driving motor system;

Step 2, the interference observer of design direct driving motor system;

Step 3, designs the continuous sliding mode controller of the finite time convergence control based on interference observer.

2. the continuous sliding-mode control of finite time of direct driving motor system interference compensation according to claim 1, is characterized in that, set up the mathematical model of direct driving motor system described in step 1, specific as follows:

Step 1-1, direct driving motor system are by being furnished with the permanent magnet DC motor Direct driver inertia load of electrical driver; According to Newton second law, the equation of motion of direct driving motor system is:

In formula (1), m is inertia load parameter, k _ifor torque error constant, B is viscosity friction coefficient, for modeling error, y is the displacement of inertia load, for the speed of inertia load, u is the control inputs of system, and t is time variable;

Step 1-2, definition status variable: then formula (1) equation of motion is converted into state equation:

In formula (2), be nominal value and known; the total interference volume of system, f (t, x ₁, x ₂) be above-mentioned x ₁represent the displacement of inertia load, x ₂represent the speed of inertia load;

Because in direct driving motor system, the state of system and parameter are all bounded, therefore the total interference volume d (x, t) of system meets:

|d(x,t)|≤D(3)

In formula (3), D is known normal number, and namely d (x, t) has the known upper bound.

3. the continuous sliding-mode control of finite time of direct driving motor system interference compensation according to claim 1, it is characterized in that, design the interference observer of direct driving motor system described in step 2, step is as follows:

Definition observer sliding-mode surface s ₁for:

s ₁＝z ₁-x ₂(4)

Wherein, z ₁for dynamic in observer;

In formula (5), k ₁, β ₁, ε ₁, p ₁and q ₁be interference observer coefficient; p ₁<q ₁, and be positive odd number, k ₁, β ₁, ε ₁be positive number, β ₁>=D;

The observer then designing the interference volume d (x, t) of direct driving motor system is:

Had by formula (2), (4), (5):

。

4. the continuous sliding-mode control of finite time of direct driving motor system interference compensation according to claim 3, it is characterized in that, the observation accuracy test process of described interference observer is:

Definition interference observer Lyapunov Equation:

Again because of β ₁>=D, 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 ₁at Finite-time convergence to equilibrium point, namely will there is a time t in (t) ₂point, at t ₂afterwards, V ₁t () perseverance is zero, by V ₁t the expression formula (9) of () is known, V ₁t () is after zero, s ₁also be zero, now also zero will be converged to, again because of d (x, t) evaluated error

The evaluated error then disturbed also will at finite time t ₂be inside 0; Namely at t ₂after have

Obtain interference observer:

5. the continuous sliding-mode control of finite time of direct driving motor system interference compensation according to claim 1, it is characterized in that, the design described in step 3 is based on the continuous sliding mode controller of the finite time convergence control of interference observer, specific as follows:

Definition direct driving motor alliance tracking error e ₀(t), error variance e ₁(t):

e ₀(t)＝x ₁-x _d(t)(14)

Wherein, x _dt () is system reference position signalling, x _dt () is Second Order Continuous, and system reference position signalling x _d(t), system reference rate signal system reference acceleration signal all bounded;

Definition sliding mode controller sliding-mode surface s:

Wherein λ ₀, λ ₁, α ₁, α ₂be sliding mode controller parameter, and be all greater than zero, and λ ₀, λ ₁meet expression formula z ²+ λ ₁z+ λ ₀be Hurwitz, wherein z is differentiating operator, α ₁, α ₂meet α ₂∈ (0,1); Then,

Obtaining sliding mode controller u is:

Wherein λ ₂, λ ₃, α ₀for controller parameter, and λ ₂>0, λ ₃>0,0< α ₀<1.

6. the continuous sliding-mode control of finite time of direct driving motor system interference compensation according to claim 5, it is characterized in that, be zero test in described direct driving motor system Existence of Global Stable and tracking error finite time, specific as follows:

Formula (18) is substituted into formula (17) have:

Definition sliding mode controller Lyapunov Equation:

Then have:

Then sliding mode controller sliding-mode surface s will be zero in finite time, namely there is a time point t ₁, at t ₁there is s=0 afterwards, from formula (16), now have:

Then have:

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

If again because there is a system such as formula shown in (24), and each parameter lambda ₀, λ ₁, α ₁, α ₂all be greater than zero, and λ ₀, λ ₁meet expression formula z ²+ λ ₁z+ λ ₀be Hurwitz, wherein z is differentiating operator, α ₁, α ₂meet α ₂∈ (0,1), then this system state e ₀(t), e ₁t () will be stabilized to equilibrium point in finite time, i.e. once-existing some t ₄, e ₀(t), e ₁t () will at t ₄converging to zero afterwards, is i.e. zero in the tracking error finite time of system.