CN106527150A - Nonlinear composite control method of pneumatic servo loading system - Google Patents

Abstract

The invention discloses a nonlinear composite control method of a pneumatic servo loading system. According to the method, the feedback linearization is used, through pole configuration, the unstable pole of the pneumatic servo loading system is eliminated, and thus the system is converged. For a defect that the dynamic uncertainty of the system and outside and robustness can be not ensured, a Lyapunov redesign method is used, and the system has robustness even a large dynamic uncertainty exists. According to the method, an ITAE optimization control method is used, and the requirements of rapidity and tracking precision of the system can be ensured at the same time.

Description

Nonlinear composite control method of pneumatic servo loading system

Technical Field

The invention relates to the field of electromechanical servo control, in particular to a nonlinear composite control method of a pneumatic servo loading system.

Background

The pneumatic system is suitable for industrial application due to the advantages of clean energy, no pollution, low cost, simplicity, easy realization, high power-weight ratio and the like, but on the other hand, the further research and application of the pneumatic servo control technology are restricted due to the characteristics of low rigidity and weak damping of gas, uncertainty of parameters of the pneumatic system, unmodeled dynamics and the like.

Disclosure of Invention

Aiming at the defects of the prior art, the invention provides a nonlinear compound control method of a pneumatic servo loading system with good dynamic performance.

In order to achieve the above object, the present invention provides a nonlinear composite control method for a pneumatic servo loading system, comprising: comprises the following steps of (a) carrying out,

1) designing a pneumatic servo loading control system;

2) establishing a system dynamics and thermodynamic equilibrium equation, and obtaining a system error expression;

3) designing a state transformation and an input transformation, and converting the system error from nonlinear control into linear steady control;

4) reducing the dynamic uncertainty existing in the system error control by utilizing a Lyapunov method;

5) and an ITAE optimization control algorithm is adopted, so that the rapidity and the tracking progress of system error control are ensured.

Further, the pneumatic servo loading control system comprises a loading channel and a loaded channel which are identical in principle and structure, the output end of the loading channel is fixedly connected with the output end of the loaded channel, the input end of the loading channel is connected to the same air source adjusting device, the loading channel and the loaded channel respectively comprise a pneumatic motor and a pneumatic valve, the air inlet end of the pneumatic valve is connected with the air source adjusting device, and the air outlet end of the pneumatic valve is connected with the pneumatic motor. Further, the specific process of step 2) is as follows:

21) under ideal assumed conditions, the flow continuity equation and the system thermodynamic equation are utilized to obtain the state equation of the two cavities of the pneumatic motor according to the mass conservation law

Wherein R represents the gas constant of air, T₁₁Is the temperature of the motor I chamber, P_sRepresenting source pressure, p₀Denotes atmospheric pressure, P₁₁Is the pressure of the motor chamber I, P₂₁Is the pressure intensity of the cavity of the motor II,representing the rate of change of mass flow, V₁₁＝D_m(θ₁₀+θ₀)，V₁₂＝D_m(θ₂₀+φ-θ₀)，S_1in，S_1outIs the effective opening area of the proportional valve air inlet; s_2in，S_2outIs the effective opening area of the intake and exhaust ports of the proportional valve, wherein S_1in＝S_2out，S_1out＝S_2in；

22) The two-cavity state equation is converted into a simulated form:

in the formula, P_iIndicating the pressure of two chambers of the pneumatic oscillating motor, u_iA control signal indicating a proportional flow valve, theta indicates a rotation angle of the motor,representing the angular velocity of the motor blades, t representing time;

23) and obtaining a moment expression generated by the pressure difference of the two cavities according to the affine form of the state equation of the two cavities:

in the formula, D_mRepresenting the displacement of the motor, u representing the control quantity input, i.e. the servo valve inputA signal;

24) converting a moment expression generated by the pressure difference of the two cavities into an imitation type form:

still further, the systematic error expression is:

in the formula, M_pRepresenting the moment produced by the pressure difference between the two chambers, M_fRepresenting the friction torque of the system; m_dIs the signal that is desired to be tracked.

Still further, the specific process of step 3) is as follows:

31) setting the state transition as:

then

Wherein,a derivative of the desired tracking signal;

32) input is changed to:

wherein v is a virtual control amount;

33) linearizing the state control:

still further, the specific process of the step 4) is,

41) the input quantity v is designed as: v ═ v_l+v_nd

Wherein, let v_l＝-KX，

In the formula, v_lDesigning feedback control variables for a linear state feedback design method, designing linear control components by adopting a pole allocation method, solving by using an LMI tool box of Matlab, and v_ndIs a nonlinear control variable;

42) the nominal Lyapunov equation is designed as:

then:

order to

In the formula,representing a friction model and associated uncertainty

43) V in 42)_ndBringing inFinishing to obtain:

for-k₁|B^TP^-1X|²||²+|η₁||B^TP^-1X | | |, whenSometimes has a maximum valueThus, it is possible to provideAt a radius ofIs always negative, the solution (X) of the closed-loop system is consistently bounded for any initial state, even in the case where the upper bound of the uncertainty dynamics is unknown, according to the provenance "Slotine J, Li]The lemma of China Machine Press,2004 ", for any initial state, even if the upper bound of the dynamic uncertainty is unknown, the solution (X) of the closed-loop system is consistently bounded, and the optimization is performed by using the ITAE optimization control algorithm, while the requirements of tracking rapidity and tracking accuracy are satisfied, which proves as follows:

for the

Let R be PQP^-1，

Then there are:

for theIs provided withAnd X^TPX≥λ_min(P)||X||²Then there isThe solution of the closed loop system is bounded consistently, namely the system is stable and can ensure certain tracking accuracy.

The invention has the advantages that:

the method adopts feedback linearization, and eliminates an unstable pole of the pneumatic servo loading system through pole allocation, so that the system is converged; aiming at the defect that the robustness of the system cannot be ensured due to the dynamic uncertainty of the system to the self and the outside, the Lyapunov redesign method is adopted to ensure that the system has the robustness even if the system has larger dynamic uncertainty. The method adopts an ITAE optimization control algorithm, and can simultaneously ensure the requirements of rapidity and tracking precision of the system.

Drawings

FIG. 1 is a pneumatic servo loading system.

FIG. 2 is a valve controlled cylinder in a pneumatic servo loading system.

FIG. 3 is a schematic diagram of a nonlinear compound control method of the pneumatic servo loading system of the present invention.

FIG. 4 is a comparison of 60 degree 20Nm0.5hz sine PID plus feedforward control and nonlinear composite control method simulation results.

FIG. 5 is a comparison of the simulation results of the 60 degree 10Nm 1Hz and PID plus feedforward control and nonlinear composite control method of the present invention.

FIG. 6 is a comparison of the experimental results of the 60 ° 20Nm0.5hz PID plus feedforward control and nonlinear compound control method of the present invention.

FIG. 7 is a comparison of the experimental results of 60 deg. 10Nm 1Hz sinusoidal opposite-vertex loading PID plus feedforward control and nonlinear control method.

Detailed Description

The invention is described in further detail below with reference to the following figures and specific examples:

a nonlinear compound control method of a pneumatic servo loading system comprises the following steps,

1) designing a pneumatic servo loading control system;

2) establishing a system dynamics and thermodynamic equilibrium equation, and obtaining a system error expression;

3) designing a state transformation and an input transformation, and converting the system error from nonlinear control into linear steady control;

4) reducing the dynamic uncertainty existing in the system error control by utilizing a Lyapunov method;

5) and an ITAE optimization control algorithm is adopted, so that the rapidity and the tracking progress of system error control are ensured.

The pneumatic servo loading control system comprises a loading channel and a loaded channel which are identical in principle and structure, the output ends of the loading channel and the loaded channel are fixedly connected, the input ends of the loading channel and the loaded channel are connected to the same air source adjusting device, the loading channel and the loaded channel respectively comprise a pneumatic motor and a pneumatic valve, the air inlet end of the pneumatic valve is connected with the air source adjusting device, and the air outlet end of the pneumatic valve is connected with the pneumatic motor. In the pneumatic servo loading system shown in fig. 1, a loading channel is arranged on the left side for performing torque servo control, a loaded channel is arranged on the right side for performing position servo control, and the two channels are fixedly connected. The whole system is designed to realize real-time sampling in actual working conditions when a loaded channel performs given position movement, and the loading channel can track and reproduce any load as much as possible. To properly simplify the mathematical model, the following assumptions are made:

a) the used working medium is ideal gas, and an ideal gas state equation is satisfied;

b) the air supply pressure P and the temperature are constant;

c) the gas in the cylinder is uniform, and the state parameters of each point in each instantaneous cavity are the same;

d) no leakage exists between the cylinder and the outside and between the two cavities;

e) the flow state of the gas flowing through the valve port or other orifices is considered as equi-entropy heat insulation process.

And establishing a system dynamics and thermodynamic equilibrium equation by the pneumatic servo loading system, and obtaining a system error expression.

In a pneumatic system, when gas flows through any component or pipeline, the gas is regarded as flowing through a small hole with a certain area, and the flow law of the gas passing through a valve port is represented by the flow characteristic of the small hole. Unlike the liquid and current characteristics, the saturation characteristics of the flow rate of gas flowing through the orifice occur with the pressure ratio between the upstream and downstream, and the mass flow rate through the orifice is given by the following equation:

in the formula Q_maIs the mass flow rate of gas flowing through the orifice; s_eIs the equivalent area of the small hole; p_uIs the pressure upstream of the orifice; p_dIs the pressure downstream of the orifice; k_GIs the gas constant; t is₁₀Is the temperature of the gas upstream of the orifice.

In fig. 2, a cell is taken around the motor I cavity, and the rate of change of mass flowing into the cell is:

the gas state equation is as follows:

unfolding to obtain:

because the piston moving speed is fast in the loading process, the gas in the motor cavity is not in time to exchange heat with the surrounding environment, the loading process is approximate to an adiabatic process, and the relationship between the gas temperature and the pressure in the process is as follows:

substituting equation (5) into equation (4) yields the following equation:

wherein q is_maMass flow through the motor cavity I; r is a gas constant; p₁₀Is the initial pressure of the motor I cavity; p₁₁Is the pressure of the motor chamber I, V₁₀Initial volume of motor I cavity; v₁₁Is the volume of the motor I cavity; t is₁₀Is the initial temperature of the motor cavity I; t is₁₁Is the temperature of the motor I cavity. From the law of conservation of mass, equation (1) is equal to (6), and the equations of state of the two chambers of the motor I, II are derived as follows:

wherein: v₁₁＝D_m(θ₁₀+θ₀)，V₁₂＝D_m(θ₂₀+φ-θ₀)，S_1in，S_1outIs the effective opening area of the proportional valve air inlet; s_2in，S_2outIs the effective opening area of the intake and exhaust ports of the proportional valve, wherein S_1in＝S_2out，S_1out＝S_2in。

The moment balance equation of the swing motor rotor of the loading system is as follows:

when a load is present:

where Δ P is the pressure difference between the two chambers of the oscillating motor, J_mTo load the system's moment of inertia, θ_mTo load angular displacement of the system, B_mTo load the viscous damping coefficient of the system, G Is torsional rigidity of torque sensor, theta_lFor load equivalent angular displacement, J_lFor load equivalent moment of inertia, B_lLoad equivalent viscous damping coefficient, G_lFor load-equivalent torsional stiffness, θ_fThe steering engine output shaft is equivalent in angular displacement.

The feedback linearization of the pneumatic servo loading system is as follows:

the following imitation patterns are expressed for (7) and (8):

wherein:

wherein:k 1.4 represents the specific heat of air, P_iThe pressure of two cavities of the pneumatic swing motor is shown; u. of_iA control voltage signal representative of a proportional flow valve; p_uIndicating the pressure upstream of the orifice; p_dIndicating the pressure downstream of the orifice; p_crRepresents the critical pressure ratio; p is a radical of₀Represents atmospheric pressure; p_sRepresenting a source pressure; r represents the gas constant of air; θ represents a rotation angle of the motor;representing the angular velocity of the motor blades; t represents the temperature of the two motor chambers during operation. The torque expression generated by the pressure difference between the two chambers is shown as formula (13):

order:

wherein M is_pThe torque generated by the pressure difference of the two cavities is shown, and delta p represents the pressure difference of the two cavities of the motor; d_mIndicating the displacement of the motor. The affine expression of the pneumatic servo loading system is obtained from the equations (13-15) as follows:

make the control voltage signal u and valve core opening area (S) of the proportional flow valve_e) Has a relationship of S_e＝K_uU | l. The system equation is further expressed as:

M_out＝M_p+M_f+M_J

wherein M is_outRepresenting the torque of the system output; m_fRepresenting the friction torque of the system; m_JRepresenting the moment of inertia of the system, which is ignored here because of the relatively small moment of inertia of the motor of the system.

For a moment servo loading system, the output expression of the system is as follows:

for a pneumatic swing motor, its own moment of inertia is much smaller than that of the load, and can be ignored. The error expression of the pneumatic servo loading system is as follows:

wherein M is_dIs the signal that is desired to be tracked.

Considering that a pneumatic system is easy to generate a creeping phenomenon in a steady state, an integral control component is introduced in the design process of a controller so as to reduce the steady-state error of the system, and the state change is designed as follows:

then:

wherein,is the derivative of the reference instruction;

the input transform is designed to:

α therein_n，β_nThe nominal values of α are expressed respectively, and the expression (17) is taken into the expression (21) to obtain:

bringing (23) into (21) yields:

by the above conversion, the problem of controlling the nonlinear system by the control amount u in the original system is converted into the problem of controlling the system (24) by the new control amount v. For a converted linear system, a pole configuration method is adopted to solve the nonlinear problem of the system, but when the system has certain dynamic uncertainty, such as parameter uncertainty, unmodeled dynamics and the like, the robustness of the system cannot be ensured by designing a controller by simply adopting a feedback linearization theory. The controller is designed as follows:

v＝v_l+v_nd(25)

for equation (25), the feedback control variable v is designed using a linear state feedback design method_l-KX, the following expression is obtained:

for equation (25), the linear control component is designed using the pole placement method, while solving through the LMI toolbox of Matlab.

Nonlinear control components and stability analysis are designed by adopting a Lyapunov redesign method as follows:

there is a unique positive definite matrix P for a given positive definite matrix Q, such that:

let the nominal system (24) Lyapunov equation be:

derivation of this equation yields:

let the expression of the nonlinear control component be:

whereinWhich represents the uncertainty in the dynamics of the system,η₁max (| η |) represents the upper bound (unknown) of the dynamic uncertainty, and bringing (31) into (30) has:

for-k₁|η||B^TP^-1X|²||²+|η||B^TP^-1X | | |, whenSometimes has a maximum valueThus, it is possible to provideAt a radius ofThe global outer is always negative, for any initial state, even under the condition that the upper bound of the dynamic uncertainty is unknown, the solution (X) of the closed-loop system is still consistent and bounded, and the ITAE optimization control algorithm is adopted to carry out real-time solutionThe requirements of tracking rapidity and tracking precision are met simultaneously, and the following is proved:

for the

Let R be PQP^-1，

Then there are:

for theIs provided withAnd X^TPX≥λ_min(P)||X||²Then there isThe solution of the closed loop system is bounded consistently, namely the system is stable and can ensure certain tracking accuracy.

The result of the simulation comparison of the PID + feedforward and nonlinear composite control method by using the simulation model is shown in the attached drawing, and FIG. 4 shows that the error range is within +/-4 Nm under the feedforward + PID control and the error is controlled within +/-2.5 Nm under the nonlinear control method under the condition that the position system is subjected to sinusoidal interference of 0.5Hz and 60 degrees, so that the system precision is obviously improved.

FIG. 5 shows a comparison of simulation results of the top loading under the condition of sinusoidal interference of 60 degrees at 1Hz in a position system, wherein the maximum amplitude of the error is within 5Nm under the control of PID + feedforward, and is within 2.5Nm under the nonlinear control method, and the maximum amplitude of the error is reduced by half; the phenomenon of creeping and buffeting is improved; while the steady state error converges quickly to near zero. Simulation results show that: the nonlinear damping control method effectively improves the control quality of the system.

Fig. 6 shows the comparison of the experimental results of the two control methods under the same conditions as fig. 5, and it can be seen from the result curves that the experimental results can be respectively matched with the corresponding simulations. The maximum error amplitude values of the corresponding experimental curves are 1Nm and 2.5Nm respectively, compared with a PID + feedforward control method, the nonlinear compound control method improves the dynamic tracking precision and the steady-state error, and improves the phenomena of crawling and buffeting. The control method can effectively improve the control quality in practice. Under experimental conditions, the error signal has relatively more buffeting, which is caused by large control parameters in the experimental process and certain noise in the sampling process.

FIG. 7 is a comparison of the experimental results of two control methods under the same conditions, and it can be seen from the result curves that the experimental results respectively correspond to the corresponding simulations, and the nonlinear composite control method is superior to the PID + feedforward control method in improving the control accuracy, the steady-state error and the phenomena of 'crawling + buffeting'. However, under experimental conditions, the error signal has a buffeting phenomenon, which is mainly caused by the existence of certain sampling noise in the experimental process.

Experimental results from several sets of data show that: the composite nonlinear composite control method of the pneumatic servo loading system can greatly improve the loading tracking precision of the system, and is better improved in the aspect of eliminating buffeting compared with a PID combined feedforward control method. Is worthy of further popularization and application.

Claims (6)

1. A nonlinear composite control method of a pneumatic servo loading system is characterized by comprising the following steps: comprises the following steps of (a) carrying out,

1) designing a pneumatic servo loading control system;

2) establishing a system dynamics and thermodynamic equilibrium equation, and obtaining a system error expression;

3) designing a state transformation and an input transformation, and converting the system error from nonlinear control into linear steady control;

4) reducing the dynamic uncertainty existing in the system error control by utilizing a Lyapunov method;

5) and an ITAE optimization control algorithm is adopted, so that the rapidity and the tracking progress of system error control are ensured.

2. The nonlinear compound control method of a pneumatic servo loading system according to claim 1, wherein: the pneumatic servo loading control system comprises a loading channel and a loaded channel, the loading channel and the loaded channel are identical in principle and structure, the output end of the loading channel is fixedly connected with the output end of the loaded channel, the input end of the loading channel is connected to the same air source adjusting device, the loading channel and the loaded channel respectively comprise a pneumatic motor and a pneumatic valve, the air inlet end of the pneumatic valve is connected with the air source adjusting device, and the air outlet end of the pneumatic valve is connected with the pneumatic motor.

3. The nonlinear compound control method of the pneumatic servo loading system according to claim 2, wherein: the specific process of the step 2) is as follows:

21) under ideal assumed conditions, the flow continuity equation and the system thermodynamic equation are utilized to obtain the state equation of the two cavities of the pneumatic motor according to the mass conservation law

${\stackrel{\·}{P}}_{11}=\frac{\∂R\sqrt{T_{11}}}{V_{11}}\[S_{1in}{P}_s\stackrel{\·}{m}(P_s,P_{11})-S_{1out}{P}_{11}\stackrel{\·}{m}(P_{11},P_0)\]-\frac{{\∂}_1{P}_{11}}{V_{11}}{\stackrel{\·}{V}}_{11}$

${\stackrel{\·}{P}}_{21}=\frac{\∂R\sqrt{T_{21}}}{V_{21}}\[S_{2in}{P}_s\stackrel{\·}{m}(P_s,P_{21})-S_{2out}{P}_{21}\stackrel{\·}{m}(P_{21},P_0)\]-\frac{{\∂}_1{P}_{21}}{V_{21}}{\stackrel{\·}{V}}_{21}$

Wherein R represents the gas constant of air, T₁₁Is the temperature of the motor I chamber, P_sRepresenting source pressure, p₀Denotes atmospheric pressure, P₁₁Is the pressure of the motor chamber I, P₂₁Is the pressure intensity of the cavity of the motor II,representing the rate of change of mass flow, V₁₁＝D_m(θ₁₀+θ₀)，V₁₂＝D_m(θ₂₀+φ-θ₀)，S_1in，S_1outIs the effective opening area of the proportional valve air inlet; s_2in，S_2outIs the effective opening area of the intake and exhaust ports of the proportional valve, wherein S_1in＝S_2out，S_1out＝S_2in；

22) The two-cavity state equation is converted into a simulated form:

${\stackrel{\·}{P}}_i=-f_i(p_{i,}\mathrm{\θ},\stackrel{\·}{\mathrm{\θ}})+g_i(t,p_i,\mathrm{\θ})u_i,i=1,2$

wherein:

$\begin{array}{ccc}{f}_i(p_{i,}\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}})=\frac{{\mathrm{kp}}_i}{V_i\left(\mathrm{\&theta;}\right)}{V}_i\left(\stackrel{\&CenterDot;}{\mathrm{\&theta;}}\right)& g_i(t,p_i,\mathrm{\&theta;})=\left\{\begin{array}{cc}\stackrel{\&CenterDot;}{m}\left(p_s,p_i\right)& u>=0\\ -\stackrel{\&CenterDot;}{m}\left(p_i,p_0\right)& u<0\end{array}\right.& u_i=\left\{\begin{array}{ccc}u& if& i=1\\ -u& if& i=2\end{array}\right.\end{array}$

$\stackrel{\&CenterDot;}{m}(P_u,P_d)=\left\{\begin{array}{cc}{K}_{G}Rk\sqrt{T}\&CenterDot;P_u& \frac{P_d}{P_u}\&le;P_{cr}\\ 2K_{G}Rk\sqrt{T}\&CenterDot;P_u\sqrt{\frac{P_d}{P_u}(1-\frac{P_d}{P_u})}& \frac{P_d}{P_u}\&GreaterEqual;P_{cr}\end{array}\right.$

wherein:k 1.4 represents the specific heat of air, P_iThe pressure of two cavities of the pneumatic swing motor is shown; u. of_iIndicating control of proportional flow valvesA voltage signal is generated; p_uIndicating the pressure upstream of the orifice; p_dIndicating the pressure downstream of the orifice; p_crRepresents the critical pressure ratio; p is a radical of₀Represents atmospheric pressure; p_sRepresenting a source pressure; r represents the gas constant of air; θ represents a rotation angle of the motor;representing the angular velocity of the motor blades; t represents the temperature of two cavities of the motor in the working process, and T represents time;

23) and obtaining a moment expression generated by the pressure difference of the two cavities according to the affine form of the state equation of the two cavities:

${\stackrel{\&CenterDot;}{M}}_p=-\&lsqb;f_1(p_1,\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}})-f_2(p_2,\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}})\&rsqb;\&CenterDot;D_m+\&lsqb;g_1(t,p_1,\mathrm{\&theta;})-g_2(t,p_2,\mathrm{\&theta;})\&rsqb;\&CenterDot;D_m\&CenterDot;u$

in the formula, D_mRepresents the displacement of the motor, u is the control quantity input, i.e. the servo valve input signal;

24) converting a moment expression generated by the pressure difference of the two cavities into an imitation type form:

${\stackrel{\&CenterDot;}{M}}_p=-f(p_1,p_2,\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}})+g(t,p_1,p_2,\mathrm{\&theta;})\&CenterDot;u.$

4. the nonlinear compound control method of the pneumatic servo loading system according to claim 3, wherein: the systematic error expression is

$e=\tilde{M}=M_p+M_f-M_d$

In the formula, M_pRepresenting the moment produced by the pressure difference between the two chambers, M_fRepresenting the friction torque of the system; m_dIs the signal that is desired to be tracked.

5. The nonlinear compound control method of the pneumatic servo loading system according to claim 4, wherein: the specific process of the step 3) is as follows:

31) setting the state transition as:

then

Wherein,a derivative of the desired tracking signal;

$\begin{array}{cc}A=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)& B=\left(\begin{array}{c}0\\ 1\end{array}\right)\end{array};$

32) input is changed to:

$u=\frac{v+f(p_1,p_2,\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}})}{g(t,p_1,p_2,\mathrm{\&theta;})}$

wherein v is a virtual control amount;

33) the state control is linearized and used as a control,

6. the nonlinear compound control method of the pneumatic servo loading system according to claim 5, wherein: the specific process of the step 4) is that,

41) the input quantity v is designed as: v ═ v_l+v_nd

Wherein, let v_l＝-KX，

In the formula, v_lDesign of feedback control variables, v, for a linear state feedback design method_ndA non-linear control variable;

42) the nominal Lyapunov equation is designed as:

then:

order to

$v_{nd}=-(k_1{{\stackrel{\&CenterDot;}{M}}_f}^2-{\stackrel{\&CenterDot;}{M}}_d)B^{T}PX+{\mathrm{\&eta;}}_1\mathrm{\&delta;}(t,\stackrel{\&CenterDot;}{\mathrm{\&theta;}},\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}})$

In the formula,representing a friction model and associated uncertainty

43) V in 42)_ndBringing inFinishing to obtain:

$\stackrel{\&CenterDot;}{V}\&le;-\frac{1}{2}{X}^T{\mathrm{PQP}}^{-1}X+X^T{\mathrm{PBv}}_{nd}+{\mathrm{\&eta;}}_1\left|X^{T}PB\mathrm{\&delta;}(t,\stackrel{\&CenterDot;}{\mathrm{\&theta;}},\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}})\right|-{\stackrel{\&CenterDot;}{M}}_d$

for theSometimes has a maximum valueThus, it is possible to provideThe solution (X) for a closed loop system is consistently bounded for any initial state, always negative outside a sphere, even if the upper bound of uncertainty dynamics is unknown, according to the provenance "Slotine J, Li W]The lemma of China Machine Press,2004 ", for any initial state, even if the upper bound of the dynamic uncertainty is unknown, the solution (X) of the closed-loop system is consistently bounded, and the dynamic uncertainty present in the system error control is reduced.