CN114296351B - Hybrid gain control method of nonlinear mechanical arm system - Google Patents

Abstract

The invention belongs to the field of control theory and control engineering, is applied to the self-adaptive control of a nonlinear system, and provides a novel mixed gain scaling method comprising static and dynamic gains for a nonlinear system with unknown continuous measurement sensitivity and uncertain nonlinearity. A hybrid gain control method of a nonlinear mechanical arm system, comprising: step 1, establishing a mathematical model describing a mechanical arm system by utilizing a dynamics principle; step 2, determining that the system is a nonlinear system, and designing a high-gain observer to estimate the state of the original system; and 3, establishing a controller to ensure that the system is stably operated.

Description

Hybrid gain control method of nonlinear mechanical arm system

Technical Field

The invention belongs to the field of control theory and control engineering, is applied to the self-adaptive control of a nonlinear system, and provides a novel mixed gain scaling method comprising static and dynamic gains for a nonlinear system with unknown continuous measurement sensitivity and uncertain nonlinearity.

Background

With the continuous development of science and technology to the intelligent direction, the application field of robots is continuously expanded and deepened, and industrial robots become a high and new technology industry, play a great role in the industrial automation level, and play an increasingly important role in future production and social development. Robots are typical representatives of advanced manufacturing techniques and automation equipment, being "final" representatives of man-made machines. The system relates to a plurality of subjects and fields of machinery, electronics, automatic control, computers, artificial intelligence, sensors, communication, networks and the like, and is the comprehensive integration of various high and new technical development achievements, so that the development of the system is closely related to the development of a plurality of subjects. After entering the 80 s of the 20 th century, with the continuous deep development of innovation and the impact of high and new technology, the development and research of the robot technology in China are valued and supported by the government, and a great deal of scientific research results are obtained after the research for several years. Robotic arms are a typical representation of industrial robots that mimic certain motion functions of a human hand and arm to grasp, handle objects or manipulate an automated device of a tool in a fixed sequence. It can replace heavy labor to realize mechanization and automation of production, and can operate in harmful environment to protect personal safety, so it can be widely used in the departments of mechanical manufacture, metallurgy, electronics, light industry and atomic energy.

Adaptive techniques are one of the effective methods of studying nonlinear systems. In daily life, by adaptation is meant a feature in which a living being can change its own habits to adapt to a new environment. Thus, intuitively, an adaptive controller should be one that can modify its own characteristics to accommodate changes in the dynamic characteristics of objects and disturbances. An adaptively controlled object of interest is a system with a degree of uncertainty, herein referred to as "uncertainty" which means that the mathematical model describing the object being controlled and its environment is not completely deterministic, including some unknown and random factors. In summary, the adaptive technology solves the uncertainty in the system through time-varying dynamic gain, and embodies the idea of dynamic braking.

The output result in the mechanical arm system cannot accurately indicate that the system state has great influence on the system, so the following design is carried out on the system, and the problem is solved.

Disclosure of Invention

The method aims to solve the problem of global self-adaptive state asymptotic adjustment of a nonlinear mechanical arm system with large uncertainty through output feedback. The invention provides a hybrid gain control method of a nonlinear mechanical arm system, which comprises the following steps:

step 1, establishing a mathematical model describing a mechanical arm system by utilizing a dynamics principle;

wherein q (t),respectively representing the angular position, speed and acceleration of the link, I (t) is the motor armature current, V _E (t) is the voltage as the control input;

wherein J is the inertia of the rotor, _m for link quality, M ₀ For loading mass, L ₀ For the link length, R ₀ Is the load radius, G is the gravity coefficient, B ₀ K is the viscous friction coefficient at the joint _τ The armature current is converted to a torque machine by a coefficient. L (L) _H ,R,K _B Is armature inductance, armature resistance and electric field coefficient, L _H M may vary with variations in rotational speed, load, temperature, etc. within an unknown range;

select state n ₁ ＝L _H Mq,n ₃ ＝L _H I, the following is obtained through variation:

wherein: θ=l _H M is the sensitivity error obtained in the calculation process;

step 2, determining that the system is a nonlinear system, and designing a high-gain observer to estimate the state of the original system;

wherein the dynamic gain L is defined as follows:

step 3, solving h in step 2 ₁ ,h ₂ ,h ₃ ,k ₁ ,k ₂ ,k ₃ ；

Make I epsilon R ^n×n Represents an identity matrix and defines matrices A, B and D as

For any constant alpha>0, always there is h _i >0,ν>0 and a numerical matrix p=p ^T So that

A ^T P+PA≤αI,DP+PD≥νI

Also always there is a set of constants beta>0,k _i >0 and numerical matrix q=q ^T So that

B ^T P+PB≤βI,DQ+QD≥0

Solving and determining h through the formula ₁ ,h ₂ ,h ₃ ,k ₁ ,k ₂ ,k ₃ ；

And 4, establishing a controller to ensure the stable operation of the system.

The invention is characterized in that: the design can reduce the influence caused by sensitivity errors in the mechanical arm system, and realize global progressive stabilization of the system.

Drawings

FIG. 1 is state n of the present invention ₁ Andis a trajectory graph of (1);

FIG. 2 is state n of the present invention ₂ Andis a trajectory graph of (1);

FIG. 3 is state n of the present invention ₃ Andis a trajectory graph of (1);

fig. 4 is a state trace diagram of the dynamic gain L and a state trace diagram of the control u of the present invention.

Detailed Description

The technical scheme of the invention is further specifically described below through specific embodiments and with reference to the accompanying drawings.

Example 1

In the present embodiment, the setting parameters are as follows.

Specific numerical values

Parameters (parameters)	J	m	R 0	M 0	L 0
						Unit (B)	kg·m 2	kg	m	kg	m
Numerical value	1.625×10 -3	0.506	0.023	0.434	0.305
						Parameters (parameters)	B 0	L H	R	K τ	K B
Unit (B)	N·m·s/rad	H	Ω	N·m/A	N·m/A
						Numerical value	16.25×10 -3	25×10 -3	5	0.9	0.9

Step 1:

a mathematical model describing the mechanical arm system is established by utilizing a dynamics principle:

q(t)，respectively represent the angular position and speed of the linkDegree and acceleration, I (t) is motor armature current, V _E And (t) is the voltage as the control input.

Wherein J is the inertia of the rotor, _m for link quality, M ₀ For loading mass, L ₀ For the link length, R ₀ Is the load radius, G is the gravity coefficient, B ₀ K is the viscous friction coefficient at the joint _τ The armature current is converted to a torque machine by a coefficient. L (L) _H ,R,K _B Is armature inductance, armature resistance and electric field coefficient, L _H M may vary over an unknown range with variations in rotational speed, load, temperature, etc.

Select state n ₁ ＝L _H Mq,n ₃ ＝L _H I, the following is obtained through variation:

wherein: θ=l _H M is the sensitivity error obtained during the calculation process, and this patent aims to eliminate its influence.

Step 2, determining that the system is a nonlinear system, and designing a high-gain observer to estimate the state of the original system;

wherein the dynamic gain L is defined as follows:

step 3, solving h in step 2 ₁ ,h ₂ ,h ₃ ,k ₁ ,k ₂ ,k ₃ ；

Make I epsilon R ^n×n Represents an identity matrix and defines matrices A, B and D as

For any constant alpha>0, always there is h _i >0,ν>0 and a numerical matrix p=p ^T So that

A ^T P+PA≤αI,DP+PD≥νI

Also always there is a set of constants beta>0,k _i >0 and numerical matrix q=q ^T So that

B ^T P+PB≤βI,DQ+QD≥0

Solving and determining h through the formula ₁ ,h ₂ ,h ₃ ,k ₁ ,k ₂ ,k ₃ ；

And 4, establishing a controller to ensure the stable operation of the system.

Next, it will be described that the system can be stably operated under the above-designed controller.

In order to facilitate the following continued research, important arguments for the proposed method of the present invention are now presented: lemma 1:

make I epsilon R ^n×n Represents an identity matrix and defines matrices A, B and D as

For any constant alpha>0, always there is h _i >0,ν>0 and a numerical matrix p=p ^T So that

A ^T P+PA≤αI,DP+PD≥νI

Also always there is a set of constants beta>0,k _i >0 and numerical matrix q=q ^T So that

B ^T P+PB≤βI,DQ+QD≥0

The lemma 2 (Barbalat lemma) is g (t) that R.fwdarw.R is a continuously differentiable function,is consistently continuous and g (t) is squared integrable, then +.>

Firstly, in order to facilitate the processing of the original system and the observer, the following coordinate transformation is performed:

note that the above transformations, the system can be rewritten as:

wherein ε= [ ε ] ₁ ε ₂ ε ₃ ] ^T ,H＝[h ₁ h ₂ h ₃ ] ^T A, B, D are given by the above-mentioned quotation marks. And is also provided with

Further, the stability of the uncertain nonlinear system is verified by using a Lyapunov function and a linear inequality method, and a positive Lyapunov function is constructed for the mechanical arm system

Based on lemma 1 pairThe derivation can be carried out:

simplifying the above formula by using the complete square formula

Wherein c ₁ ,c ₂ Two unknown constants. m is m ₁ ,m ₂ Is two known constants

And then can obtain:

the original system was then analyzed for various state stabilities of the observer and controller based on the derivative of the lyapunov function.

Step 3.1: proof of L being bounded

Let L be at [0, t _f ) The above is unbounded. So that the number of the parts to be processed,

wherein the time of existence t ₁ ，t ₁ ≤t≤t _f So that

γ ₂ L-c ₂ ≥1,γ ₂ L-c ₂ ≥1

Can be obtained by combining the above

By passing throughCan be obtained by definition of (2)

As a result, it was obtainedIf the value is smaller than a certain constant, the value contradicts the set L, so that L can be obtained.

Step 3.2: it is proved that n,tending towards zero; based on the setting L being bounded, L is set to [0, t _f ) Upper limit is provided with->So L (t) _f ) Not less than L, and defining a sufficiently large constant L ^* The following conditions are satisfied

Wherein the method comprises the steps ofFrom subsequent calculations.

For i=1, 2, 3, a new scaling change is introduced:

note that the above transformations, the system can be rewritten as:

wherein:

b＝[0 0 1] ^T ,/>

and is also provided withThe following conditions are satisfied by the lemma decision

A ^*T P ^* +P ^* A ^* ≤-2I

Then the lyapunov function is set for the above system as follows

The derivation of the above can be obtained

By combining all the above expressions with the complete square formula, the following expressions can be obtained through a series of simplifications

Combining the definition of L with the above formula can be obtained

So it can be further simplified to obtain

Wherein gamma (L) ^* ) Is independent of L ^* Is a positive constant of (c). Then, the following formula can be obtained by integrating both sides of the above formula.

Then, the following formula can be obtained by integrating both sides of the above formula.

Because of V ^* We conclude that all closed loop states are bounded, hence t _f = + infinity of the two points, in addition due to epsilon ^* 、Is bounded and->We can get this by using the Barbalat theorem

Then utilize p epsilon ^* Anddefinition of (c) can give n _i And->Will gradually approach zero over time.

The exact expression of the nonlinear system of the mechanical arm can be obtained by utilizing parameters in the table, and the initial condition of the system is assumed to beSelecting an appropriate design parameter h ₁ ＝0.3,h ₂ ＝0.55,h ₃ ＝0.275；k ₁ ＝0.05,k ₂ ＝0.3,k ₃ ＝0.5；ι＝3；/>Simulation with matlab at σ=0.25 can yield the results shown in fig. 1, 2, 3 and 4.

Claims (1)

1. A hybrid gain control method for a nonlinear mechanical arm system, comprising:

step 1, establishing a mathematical model describing a mechanical arm system by utilizing a dynamics principle;

wherein q (t),respectively representing the angular position, speed and acceleration of the link, I (t) is the motor armature current, V _E (t) is the voltage as the control input;

wherein J is rotor inertia, M is link mass, M ₀ For loading mass, L ₀ For the link length, R ₀ Is the load radius, G is the gravity coefficient, B ₀ K is the viscous friction coefficient at the joint _τ For converting armature current into torque machine conversion coefficient, L _H ,R,K _B Is armature inductance, armature resistance and electric field coefficient, L _H M may vary with variations in rotational speed, load, temperature, etc. within an unknown range;

selecting a staten ₃ ＝L _H I, the following is obtained through variation:

wherein: θ=l _H M is the sensitivity error obtained in the calculation process;

step 2, determining that the system is a nonlinear system, and designing a high-gain observer to estimate the state of the original system;

wherein the dynamic gain L is defined as follows:

step 3, solving h in step 2 ₁ ,h ₂ ,h ₃ ,k ₁ ,k ₂ ,k ₃ ；

Make I epsilon R ^n×n Represents an identity matrix and defines matrices A, B and D as

For any constant alpha>0, always there is h _i >0,ν>0 and a numerical matrix p=p ^T So that

A ^T P+PA≤αI,DP+PD≥νI

Also always there is a set of constants beta>0,k _i >0 and numerical matrix q=q ^T So that

B ^T P+PB≤βI,DQ+QD≥0

Solving and determining h through the formula ₁ ,h ₂ ,h ₃ ,k ₁ ,k ₂ ,k ₃ ；

And 4, establishing a controller to ensure the stable operation of the system.