CN116500908B - Mechanical arm anti-interference constraint control method of double-disturbance observer - Google Patents
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Abstract
The invention discloses a mechanical arm anti-interference constraint control method of a double-disturbance observer, which comprises the following steps: 1) Establishing a dynamics model of a single-joint mechanical arm system; 2) Designing an exogenous interference modeling and double-interference observer; 3) Analyzing the stability of the double-interference observer; 4) Constructing a logarithmic barrier Liziplov function, and designing an inversion controller according to the state tracking error and the virtual control law; 5) Verifying the Lyapunov stability of the whole closed loop system; 6) And (3) carrying out simulation verification through Matlab, and judging whether readjustment of parameters is needed or not according to the simulation effect. The single-joint mechanical arm state constraint anti-disturbance control method provided by the invention realizes that the single-joint mechanical arm can quickly recover track tracking when being subjected to multi-source disturbance by utilizing the double-disturbance observer and the inversion method based on the barrier Lyapunov, and ensures that the measurable output of the system and all state variables do not violate constraint boundaries.
Description
Technical Field
The invention relates to the technical field of automatic control, in particular to an anti-interference constraint control method for a mechanical arm of a double-disturbance observer.
Background
Along with the rapid development of the industrialization process in China, the traditional manufacturing industry and the high and new technology industry put higher requirements on the control precision and performance of the mechanical arm. Compared with the multi-joint mechanical arm, the single-joint mechanical arm has the characteristics of smaller volume, lighter weight, high load-weight ratio, low damping and the like, and is widely applied to the occasions such as aerospace operation, biomedical field, modern industrial manufacturing and the like. However, since a single-joint mechanical arm can be regarded as a complex nonlinear system, there is a strong coupling characteristic with multiple inputs and multiple outputs. In addition, the existence of the mechanical arm joint and external interference can cause different degrees of vibration of the mechanical arm system, which not only reduces the control performance of the mechanical arm system, but also can cause the system to run away. The above problem has become a "bottleneck" problem of arm link angle control, which often results in that the actuator of the single-joint arm is in a low-precision control state for a long time when performing the track following operation, so that it is needed to design a more effective anti-interference robust control algorithm to change the current situation.
Adaptive fuzzy control is often adopted in anti-disturbance control, but the method only carries out simple fuzzy processing on information, which leads to the reduction of control precision of a system and the failure of a mechanical arm system such as: when the clamping knife, the band-type brake and the sensor are in fault, misoperation is easy to occur. Therefore, simple adaptive control fuzzy control cannot be used in a high safety environment. In recent years, as an active anti-disturbance control method, namely a disturbance observer-based compensation method, the method has the advantages of simple structure, low cost, strong compensation capability, strong portability and the like, and becomes a main development direction of the anti-disturbance control of the mechanical arm. However, since the mechanical arm system has many unmodeled disturbances of uncertain amplitude and frequency in practical applications, it has a certain influence on the robustness of the control method.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides an anti-interference constraint control method for a mechanical arm of a double-disturbance observer, which adopts an inversion algorithm and introduces a barrier Liziplov function to ensure that the angular speed, the angular acceleration and the current of a driving motor of a connecting rod of a single-joint mechanical arm are practically constrained; then, by designing the double-interference observer and adding an interference compensation term into the controller, the robustness of external interference is improved, the single-joint mechanical arm can quickly recover stable track tracking, and adverse effects of external interference on the mechanical arm are restrained.
The purpose of the invention is realized in the following way: a mechanical arm anti-interference constraint control method of a double-disturbance observer comprises the following steps:
step 1), establishing a dynamic model of a single-joint mechanical arm system, and finishing initializing a system and updating control parameters;
step 2) designing a double-interference observer aiming at exogenous interference and unknown unmodeled interference according to a state equation equivalent to the system, and compensating in a controller;
step 3) constructing a Liapunov function, and analyzing the stability of the double-interference observer;
step 4) designing a control law by adopting an inversion method and a logarithmic barrier Liapunov function according to model information of the single-joint mechanical arm system; constructing a logarithmic barrier Liziplov function, and designing an inversion control law according to the state tracking error and the virtual control law;
step 5) stability analysis of the control law, verifying the Lyapunov stability of the whole closed-loop system;
and 6) carrying out simulation verification through Matlab, and judging whether readjustment of parameters is needed or not according to the simulation effect.
As a further definition of the present invention, the step 1) specifically includes: analyzing and modeling a single-joint mechanical arm system; according to a kinetic equation, analyzing the single-joint mechanical arm system to obtain a mathematical model as follows:
wherein,q is the link angle of the mechanical arm, +.>Is the angular velocity of the connecting rod>Is the angular acceleration of the connecting rod; i represents the armature current of the driving motor, and V is the input control voltage; j represents rotor inertia, B is viscosity friction coefficient, m is weight g of the connecting rod and is gravity acceleration; m is M l Representing load mass, L l Is the length of the connecting rod, R l Is the load radius; k (K) c Representing the conversion coefficient, K B Is the back electromotive force coefficient L a Is armature inductance, R a Is a resistor; d, d 2 (t) and d 3 (t) is modelable exogenous interference, φ 3 (t) is a derivative-bounded non-modelable disturbance;
initializing system state parameters: defining a state variable x 1 ＝q,x 3 I, the output of the system is y=x 1 The method comprises the steps of carrying out a first treatment on the surface of the The single joint mechanical arm system model of formula (1) is rewritten as follows:
wherein,
as a further definition of the present invention, the step 2) specifically includes:
suppose exogenous interference d 2 (t) and d 3 (t) can be generated by the following exogenous system:
wherein h is i (t)∈R r Is a state variable of an exogenous system, T i ∈R 1×r ,Ξ i ∈R r×r Is a known dimensionality matrix, and (xi i ,T i ) Is observable; the foreign system (3) represents a harmonic disturbance of known frequency, but of unknown phase and amplitude;
for exogenous interference d 2 (t) and d 3 (t) designing a modelable disturbance observer to be:
wherein x is 4 ＝u,L i Is the observer gain ζ i Is an auxiliary vector as an observer. />And->Respectively is h i (t),d i An estimate of (t);
defining the estimation error epsilon of an observer i (t) is:
from equation (5), the dynamic equation for the estimated error can be obtained as:
wherein,estimating errors for observer, i.e. +.>
For interference phi 3 (t) introducing an auxiliary variable:
θ 3 (t)＝φ 3 (t)-Θ 3 x 3 (7)
as can be seen from the formula (6),
wherein,is external interference phi 3 Observations of (t), Θ 3 Is a positive parameter;
the nonlinear disturbance observer is designed to be:
from equation (9), the dynamic equation for the observer estimation error can be:
wherein,estimating errors for observer, i.e. +.>
Thus, from equations (4) and (10), the double interference observer is obtained as:
as a further definition of the invention, said step 3) comprises in particular the construction of a li-apunov function:
deriving formula (9) and obtaining according to the young's inequality:
wherein phi is 3 (t) satisfies the inequality:is a positive constant.
As a further definition of the present invention, the step 4) specifically includes: dividing a third-order system into three subsystems, and designing a virtual control law for each system so that the ground virtual feedback control law of each subsystem can calm the subsystem; introducing an error variable and constructing a logarithmic type Liapunov function; and calculating an inversion control law according to the tracking error, the virtual control law and the estimated value obtained by the double-interference observer, wherein the inversion control law is shown as a formula (14):
wherein k is 3 Is a constant that is greater than zero and,
as a further definition of the present invention, the step 5) specifically includes: taking the Lyapunov function of the closed loop system as V 3 And the following inequality (15) is established, the virtual control law, the actual control law and the disturbance observer can verify and obtainTherefore, all signals of the closed loop system are guaranteed to be consistent and terminated in a bounded mode, the output and all states of the system are in corresponding constraint, and the estimation error of the interference observer can be effectively converged into a small field near the origin;
wherein I is an identity matrix, κ 2 And kappa (kappa) 3 Is a positive constant.
The invention adopts the technical proposal, and has the beneficial effects compared with the prior art that: (1) Aiming at adverse effects caused by unknown disturbance in a single-joint mechanical arm system, the invention designs the double-disturbance observer, and can effectively compensate both exogenous disturbance and unknown non-modeling disturbance. In addition, the L1 performance index is used to optimize the impact of system disturbances on the measurable output.
(2) The logarithmic barrier Liapunov function is constructed, so that all state variables are guaranteed not to violate function constraint under the condition that the system is disturbed, track tracking can be quickly recovered, robust control of the single-joint mechanical arm system is effectively realized, influence of external interference on the system is reduced, and control precision of the system is improved.
Drawings
FIG. 1 is a flow chart of the steps of the present invention.
Fig. 2 is a schematic view of the joint mechanical arm according to the present invention.
FIG. 3 is a schematic diagram of a closed loop control system according to the present invention.
FIG. 4 is a schematic diagram of a link angle and angular velocity tracking simulation of the present invention; (a) is a simulation diagram of the angle tracking of the connecting rod; and (b) is a simulation schematic diagram of the angular velocity tracking of the connecting rod.
FIG. 5 is a schematic diagram of a control input simulation of the present invention.
Fig. 6 is a schematic diagram of external interference and interference estimation simulation according to the present invention.
Detailed Description
The method for controlling the anti-interference constraint of the mechanical arm of the double-disturbance observer shown in fig. 1 comprises the following steps:
step 1), establishing a dynamic model of a single-joint mechanical arm system, and finishing initializing a system and updating control parameters;
establishing a dynamics model of the single-joint mechanical arm system:
wherein,q,/>the connecting rod angle, the connecting rod angular velocity and the connecting rod angular acceleration of the mechanical arm are respectively. I denotes an armature current of the drive motor, and V is an input control voltage. J. B, m and g respectively represent rotor inertia, viscous friction coefficient, mass of the connecting rod and gravitational acceleration. M is M l 、L l 、R l Respectively representing load mass, connecting rod length and load radius. K (K) c 、K B 、L a 、R a Respectively representing the conversion coefficient, back emf coefficient, armature inductance and resistance. d, d 2 (t) and d 3 (t) is modelable exogenous interference, φ 3 (t) is a derivative bounded unmodeled disturbance.
The control target of the system is to ensure that all state quantities of the single-joint mechanical arm system are constrained by an actual function under the condition that the single-joint mechanical arm system is subjected to external interference, and the connecting rod angle q of the system can track the set expected angle q d 。
Initializing system state parameters: defining a state variable x 1 ＝q,x 3 I, the output of the system is y=x 1 The method comprises the steps of carrying out a first treatment on the surface of the The single joint mechanical arm system model of formula (1) can be rewritten as follows:
wherein,
step 2) designing a double-interference observer aiming at exogenous interference and unknown unmodeled interference according to a state equation equivalent to the system, and compensating in a controller;
modeling of exogenous interference and design of double-interference observer, and exogenous interference d is assumed 2 (t) and d 3 (t) can be generated by the following exogenous system:
wherein h is i (t)∈R r Is a state variable of an exogenous system, T i ∈R 1×r ,Ξ i ∈R r×r Is a known dimensionality matrix, and (xi i ,T i ) Is observable; the foreign system (3) may represent a harmonic disturbance of known frequency (if known), but of unknown phase and amplitude.
For exogenous interference d 2 (t) and d 3 (t) designing a modelable disturbance observer to be:
wherein x is 4 ＝u,L i Is the observer gain ζ i Is an auxiliary vector as an observer. />And->Respectively is h i (t),d i An estimate of (t).
Defining the estimation error epsilon of an observer i (t) is:
from equation (5), the dynamic equation for the estimated error can be obtained as:
wherein,estimating errors for observer, i.e. +.>
For interference phi 3 (t) introducing an auxiliary variable:
θ 3 (t)＝φ 3 (t)-Θ 3 x 3 (7)
as can be seen from the formula (6),
wherein,is external interference phi 3 Observations of (t), Θ 3 Is a positive parameter.
Further, the nonlinear disturbance observer is designed to be:
from equation (9), the dynamic equation for the observer estimation error can be:
wherein,estimating errors for observer, i.e. +.>
Thus, from equations (4) and (10), we can obtain the double interference observer as:
step 3) stability analysis of double interference observer
Constructing a Liapunov function:
deriving the formula (12) to obtain:
wherein phi is 3 (t) satisfies the inequality:is a positive constant.
Step 4) designing a control law by adopting an inversion method and a logarithmic barrier Liapunov function according to model information of the single-joint mechanical arm system; constructing a logarithmic barrier Liziplov function, and designing an inversion control law according to the state tracking error and the virtual control law;
first, the third-order system is divided into three subsystems, and a virtual control law is designed for each system, so that the ground virtual feedback control rate of each subsystem can calm the subsystem. Then, error variables are introduced and a logarithmic type of Leidefenov function is constructed. As long as all errors are asymptotically converged to zero and the logarithmic barrier Liapunov function is guaranteed to remain bounded in the closed loop system, each subsystem can be guaranteed to be asymptotically stable under the action of a virtual control law and the target variable is enabled to be in a constraint range. Finally, in the third step, the Lyapunov function of the whole system is rounded and negatively set, so as to obtain the actual control input u of the system.
The inversion control law is designed as follows:
from step 1), the single-joint mechanical arm connecting rod angle q is defined as x 1 The tracked desired link angleDegree q d Defined as y d The tracking error can be defined as z 1 ＝x 1 -y d As can be seen from the formula (2), the error equation is:
wherein z is 2 ＝x 2 -α 1 ，α 1 Is a virtual control law of the first subsystem.
Constructing a lyapunov function of the first subsystem:
wherein ζ is a positive integer, m a1 (y d T) is the tracking error z 1 Is defined in the specification.
For equation (15), the derivative is obtained:
wherein,
the virtual control law alpha can be designed according to the Young's inequality and the formula (16) 1 The method comprises the following steps:
wherein k is 1 Is a constant that is greater than zero and,
substitution of formula (17) into formula (16) yields:
defining an error variable z 2 The method comprises the following steps:
z 2 ＝x 2 -α 1 (19)
and deriving the following steps:
wherein z is 3 ＝x 3 -α 2 ，α 2 Is a virtual control law of the second subsystem.
Deriving the formula (17) to obtain:
constructing a second lispro function:
wherein m is a2 (x 1 T) is the tracking error z 2 Is defined in the specification.
Deriving the formula (22) to obtain:
wherein,
the virtual control law alpha can be designed according to the formula (23) 2 The method comprises the following steps:
wherein k is 2 Is a constant that is greater than zero and,
substituting formula (24) into formula (23) yields:
defining an error variable z 3 The method comprises the following steps:
z 3 ＝x 3 -α 2 (26)
for z 3 Deriving from formula (2):
for alpha 2 And (3) obtaining a partial derivative, namely:
constructing a third lispro function as:
for V 3 Derivative is obtained by:
wherein,
the control law is thus designed as follows:
wherein k is 3 Is a constant that is greater than zero and,
substituting formula (31) into formula (30) yields:
step 5) stability analysis of control laws
The Lyapunov stability of the whole closed-loop system is verified, and the Lyapunov function of the closed-loop system is taken as V 3 And the following inequality (33) is established, the verification can be obtained under the action of the virtual control laws (17), (24), the actual control laws (31) and the disturbance observer (11)Thereby ensuring that all signals of the closed loop system are consistent and terminated in a bounded manner, that the output and all states of the system are within corresponding constraints, and that the estimation error of the disturbance observer can effectively converge to a small area near the origin.
Wherein I is an identity matrix, κ 2 And kappa (kappa) 3 Is a positive constant.
And (3) proving:
substituting equation (33) into equation (32), equation (32) may be further expressed as:
wherein, the expressions of alpha and gamma are respectively:
selecting parameters k 1 ＞0,k 2 ＞0.5,k 3 ＞0,κ 2 ＞0,κ 3 ＞0,η 2 ＞0.5,η 3 > 0.5, α > 0.
Integrating two sides simultaneously to obtain:
formula (35) shows that V is at t.fwdarw.infinity 3 (t) is bounded, satisfyThus, it is apparent that z can be obtained 1 ,z 2 ,z 3 ,α 1 ,α 2 ,ε 2 ,ε 3 ,/>Are all bounded. Since the desired trajectory is bounded by its respective derivatives, it is possible to obtain from the quotients: -m a1 (t)+y d (t)≤y(t)≤m a1 (t)+y d (t), definitionm b1 (t)＝-m a1 (t)+y d (t),/>The system output signal +.>In addition, virtual control law alpha 1 ,α 2 Dependent on->So that alpha can be obtained 1 ，α 2 Are all bounded, i.e.)>And because of x 2 ≤|z 2 |+|α 1 |,x 3 ≤|z 3 |+|α 2 I, can get:wherein,m b2 (t),/> m b3 (t),/>the expression of (2) is as follows:
thus, it is demonstrated that all closed loop signals of the system can remain consistently terminated, bounded and stable, and that all states do not violate constraints.
Step 6) in order to better embody the superiority and effectiveness of the control method provided herein, the single-joint mechanical arm system is subjected to algorithm simulation;
in the simulation experiment, the expected joint angle is set to sin pi (t, the initial value of the system is set to x 1 ＝0.3x 2 ，＝0.x1 3 -parameters in exogenous interference model (3) are selected as:
the selected unknown unmovable perturbation is 0.5sign (x 2 )+2x 2 +3cos (0.3pi.t), design the adjustable parameter of the virtual control law and the controller as k 1 ＝2，k 2 ＝2，k 3 ＝2，κ 2 ＝1，κ 3 ＝1，η 2 ＝4，η 3 > 0.5. The constraint functions that the system output and state need to satisfy are as follows:
the control system diagram and the closed loop system control block diagram are shown in fig. 2-3, the simulation effect is shown in fig. 4-6, in fig. 4, (a) is a graph of the single-joint mechanical arm joint angle tracking effect, and in fig. 4, (b) is a graph of the single-joint mechanical arm joint angular velocity tracking effectEffect diagram, y d Andin order to make the track of the object desired,m b1 (t)，/>the boundary is constrained as a function of the joint angle,m b2 (t)，/>the boundary is constrained as a function of the angular velocity of the joint. FIG. 5 is an illustration of external disturbances and their estimates experienced by a single joint robotic arm system. Fig. 6 is a control input signal for a single joint robotic arm system. Therefore, by adopting the method, the high-precision tracking of the expected angle and the angular speed can be realized on the premise of ensuring that all state variables do not violate constraint, and the influence of external interference is effectively overcome.
The invention is not limited to the above embodiments, and based on the technical solution disclosed in the invention, a person skilled in the art may make some substitutions and modifications to some technical features thereof without creative effort according to the technical content disclosed, and all the substitutions and modifications are within the protection scope of the invention.
Claims (5)
1. The mechanical arm anti-interference constraint control method of the double-disturbance observer is characterized by comprising the following steps of:
step 1), establishing a dynamic model of a single-joint mechanical arm system, and finishing initializing a system and updating control parameters;
step 2) designing a double-interference observer aiming at exogenous interference and unknown unmodeled interference according to a state equation equivalent to the system, and compensating in a controller;
the step 2) specifically comprises the following steps:
suppose exogenous interference d 2 (t) and d 3 (t) can be generated by the following exogenous system:
wherein h is i (t)∈R r Is a state variable of an exogenous system, T i ∈R 1×r ,Ξ i ∈R r×r Is a known dimensionality matrix, and (xi i ,T i ) Is observable; the foreign system (3) represents a harmonic disturbance of known frequency, but of unknown phase and amplitude;
for exogenous interference d 2 (t) and d 3 (t) designing a modelable disturbance observer to be:
wherein,L i is the observer gain ζ i Is an auxiliary vector as an observer; />And->Respectively is h i (t),d i An estimate of (t);
defining the estimation error epsilon of an observer i (t) is:
from equation (5), the dynamic equation for the estimated error is:
wherein,estimating errors for observer, i.e. +.>
For interference phi 3 (t) introducing an auxiliary variable:
θ 3 (t)＝φ 3 (t)-Θ 3 x 3 (7)
as can be seen from the formula (6),
wherein,is external interference phi 3 Observations of (t), Θ 3 Is a positive parameter;
the nonlinear disturbance observer is designed to be:
from equation (9), the dynamic equation for the observer estimation error is:
wherein,estimating errors for observer, i.e. +.>
Thus, from equations (4) and (10), the double interference observer is obtained as:
step 3) constructing a Liapunov function, and analyzing the stability of the double-interference observer;
step 4) designing a control law by adopting an inversion method and a logarithmic barrier Liapunov function according to model information of the single-joint mechanical arm system; constructing a logarithmic barrier Liziplov function, and designing an inversion control law according to the state tracking error and the virtual control law;
step 5) stability analysis of the control law, and verification of the Liapunov stability of the whole closed-loop system;
and 6) carrying out simulation verification through Matlab, and judging whether readjustment of parameters is needed or not according to the simulation effect.
2. The method for controlling the anti-interference constraint of the mechanical arm of the double-disturbance observer according to claim 1, wherein the step 1) specifically includes: analyzing and modeling a single-joint mechanical arm system; according to a kinetic equation, analyzing the single-joint mechanical arm system to obtain a mathematical model as follows:
wherein,q is the link angle of the mechanical arm, +.>Is the angular velocity of the connecting rod>Is the angular acceleration of the connecting rod; i represents armature electricity of driving motorFlow, V is the input control voltage; j represents rotor inertia, B is viscosity friction coefficient, m is weight g of the connecting rod and is gravity acceleration; m is M l Representing load mass, L l Is the length of the connecting rod, R l Is the load radius; k (K) c Representing the conversion coefficient, K B Is the back electromotive force coefficient L a Is armature inductance, R a Is a resistor; d, d 2 (t) and d 3 (t) is modelable exogenous interference, φ 3 (t) is a derivative-bounded non-modelable disturbance;
initializing system state parameters: defining a state variable x 1 ＝q,x 3 I, the output of the system is y=x 1 The method comprises the steps of carrying out a first treatment on the surface of the The single joint mechanical arm system model of formula (1) is rewritten as follows:
wherein,
3. the method for controlling the anti-interference constraint of the mechanical arm of the double-disturbance observer according to claim 1, wherein the step 3) specifically includes the steps of constructing a lisapunov function:
deriving formula (12) and deriving from the young's inequality:
wherein phi is 3 (t) satisfyInequality: is a positive constant.
4. The method for controlling the anti-interference constraint of the mechanical arm of the double-disturbance observer according to claim 1, wherein the step 4) specifically comprises: dividing a third-order system into three subsystems, and designing a virtual control law for each system so that the ground virtual feedback control law of each subsystem can calm the subsystem; introducing an error variable and constructing a logarithmic type Liapunov function; and calculating an inversion control law according to the tracking error, the virtual control law and the estimated value obtained by the double-interference observer, wherein the inversion control law is shown as a formula (14):
wherein k is 3 Is a constant that is greater than zero and,
5. the method for controlling the anti-interference constraint of the mechanical arm of the double-disturbance observer according to claim 1, wherein the step 5) specifically comprises: taking the Lyapunov function of the closed loop system as V 3 And the following inequality (15) is established, the virtual control law, the actual control law and the disturbance observer can verify and obtainTherefore, all signals of the closed loop system are guaranteed to be consistent and terminated in a bounded mode, the output and all states of the system are in corresponding constraint, and the estimation error of the interference observer can be effectively converged into a small field near the origin;
wherein I is an identity matrix, κ 2 And kappa (kappa) 3 Is a positive constant.
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