CN114310894B - Measurement output feedback control method of fourth-order uncertain nonlinear mechanical arm system - Google Patents

Abstract

The invention discloses a measurement output feedback control method of a fourth-order uncertain nonlinear manipulator system. The invention only uses the lower bound information of time-varying parameters to explicitly construct a pair of solutions of uncertain matrix inequalities. Then, these matrix inequalities are combined with an improved dynamic scaling technique to solve the problem of globally adaptive state asymptotic adjustment of the manipulator system through output feedback.

Description

Measurement output feedback control method for fourth-order uncertain nonlinear manipulator system

technical field

This paper discloses a new hybrid gain scaling method including static and dynamic gains for a manipulator system with unknown continuous measurement sensitivity and uncertain nonlinearity. First, solutions to a pair of uncertain matrix inequalities are explicitly constructed using only the lower bound information of the time-varying parameters. Then, these matrix inequalities are combined with an improved dynamic scaling technique to solve the problem of globally adaptive state asymptotic adjustment of the manipulator system through output feedback.

Background technique

Based on the continuous innovation of science and technology and the development towards intelligence, the application of robots has been continuously broadened and deepened. The application of robots to industry has become the development trend and mainstream, and will play an increasingly important role in future production and social development. Robot is a typical representative of advanced manufacturing technology and automation equipment, and the "ultimate" representative of man-made machines. Robot involves many disciplines and fields such as machinery, electronics, automatic control, computer, artificial intelligence, sensor, communication and network, etc. It is a comprehensive integration of various high-tech development achievements, so its development is closely related to the development of many disciplines. In recent years, with the continuous development of science and technology, the country has paid more and more attention to this aspect. The development and research of robot technology in my country has received the attention and support of the government. After continuous and in-depth research by Chinese researchers, a large number of scientific research results have been achieved. . The mechanical arm is a typical representative of industrial robots. It can imitate certain action functions of human hands and arms. It can replace human heavy labor to realize mechanization and automation of production, and can operate in harmful environments to protect personal safety. Therefore, it is widely used Used in electronics, light industry and atomic energy and other departments.

Contents of the invention

Aiming at the deficiencies of the prior art, the invention proposes a measurement output feedback control method of a fourth-order uncertain nonlinear manipulator system.

Step 1: Analyze the robotic arm system and establish a corresponding system model.

Step 2: Analyze the system and establish corresponding observers;

Step 3: Use dynamic scaling technology, Lyapunov function and linear matrix inequality to deduce that the state of the system is bounded and the output is ultimately kept within a preset range.

Step 4: Simulation verification results.

Compared with the existing technology, the present invention has the effect that the manipulator system has been widely used in industrial production, and this patent solves the asymptotic global adaptive state of the output feedback of a nonlinear manipulator system with large measurement uncertainty Regulatory issues. At the same time, the proposed control method is generalized to more general nonlinear manipulator systems with polynomial growth conditions of unknown parameters. Notably, this paper proposes a non-backward design approach where all design parameters are easily determined by a set of explicit constraints.

Description of drawings

Figure 1 shows the actual x ₁ , x ₂ , and observed values The state response curve of

Figure 2 shows the actual x ₃ , x ₄ , and observed values The state response curve of

Figure 3 is the state response curve of L and u.

Detailed ways

The present invention is a measurement output feedback control method of a fourth-order uncertain nonlinear manipulator system. The method specifically includes the following steps:

step one:

The dynamic model of the manipulator system is as follows:

As shown above, the unknown q ₁ represents the displacement of the connecting rod in the system, q ₂ is equal to the displacement of the rotor, q1 represents the displacement of the connecting rod, J ₁ represents the inertia of the connecting rod, J _m is equal to the inertia of the motor rotor, k ₀ represents the elastic constant, g is the gravitational constant, m is equal to the mass, l ₀ represents the center of mass, F ₁ represents the viscous friction coefficient of the connecting rod, F _m represents the viscous friction coefficient of the motor rotor, u is the torque transmitted by the motor, and only q ₁ is measurable among these variables .

The specific parameters of the robotic arm are shown below:

Table 1 Parameters of the robotic arm system

The state space model of a fourth-order manipulator system is shown below:

y = θ(t) x ₁

The following is a model with specific parameters:

y＝|1+2sin10t|x ₁

There are some errors in the actual measurement, so the concept of sensitivity θ(t) is introduced here, and it is assumed that the sensitivity θ(t) is continuous and meets 0≤θ ₁ ≤θ(t)≤θ ₂ , where θ ₁ , θ ₂ is a known constant.

Step two:

Two lemmas are introduced below, one of which is a new lemma that has not been used in the design.

First, an identity matrix is represented by I∈R ^4×4 , and then the matrices A,B,D are defined.

θ(t) is the unknown measurement sensitivity, the lower bound is a constant θ ₁ , σ>0 is also a constant, h _i >0, ki _> 0 are design degrees of freedom.

Lemma 1: For any constant α>0, there are constants h _i >0, V>0, and a digital matrix such that P=P ^T >0 so that A ^T P+PA≤-αI, DP+PD≥VI;

Lemma 2: For any constant α>0, there are constants _ki >0, β>0, and a digital matrix such that Q=Q ^T >0 so that B ^T Q+QB≤-βI,DQ+QD＞0.

Suppose i=1,2,3...n satisfy the linear growth condition: |f _i (t,x,v)|≤c(|x ₁ |+.......+|x _n |), where c >0, is one

An unknown constant, which is called the unknown growth rate; according to the parameters obtained in Lemma 1 and 2 and design a dynamic output

According to the above-mentioned manipulator system, the standard form of the fourth-order system observer is designed as follows:

L(0)=1

L is mainly determined by the above formula, where σ, is a design constant and /> 0<σ<0.5,τ>1 where design parameters; h ₁ =0.3; h ₂ =1.8; h ₃ =0.3; h ₄ =0.8; k ₁ =0.6; k ₂ =1.5; k ₃ =1.7; σ = 0.45 />

Step three:

When i=1,2,3,4, make Then introduce a scaling transformation:

According to the above scaling transformation and The system is described as:

where ε＝[ε ₁ ,ε ₂ ,ε ₃ ,ε ₄ ] ^T , H＝[h ₁ ,h ₂ ,h ₃ ,h ₄ ] ^T

f ₁ (t,v,x)=0

f ₃ (t,v,x)=0

For the system assumed above, under the condition that the dynamic gain is bounded and other closed-loop states converge to zero globally, the global self-adaptive state adjustment can be realized through the control scheme constituted above.

First choose a Lyapunov function:

Depend on Deduced:

When 0<σ<0.5, the design parameter τ is found to satisfy:

0.5α-m ₁ τ ^-2σ ≥ γ ₁

βτ-kτ ^2σ -m ₂ ≥γ ₂

γ ₁ and γ ₂ are suitable constants.

So from:

γ ₁ Lc ₁ , γ ₂ Lc ₂ may be negative numbers. Therefore, it is not a standard Lyapunov function, that is, the asymptotic stability of the closed-loop system cannot be directly proved according to the Lyapunov stability theorem.

Suppose the solution has a maximum interval [0,t _f ), t _f ∈(0,+∞) or t _f =+∞.

Making L(t) bounded in [0,t _f ) will prove the asymptotic convergence of the asymptotic system.

Prove that L is bounded:

Assume L is unbounded on [0, t _f ). so,

where there exists time t ₁ , t ₁ ≤t≤t _f , such that

Υ ₂ Lc ₂ ≥ 1, Υ ₂ Lc ₂ ≥ 1

Combined with the above formula, we can get

pass definition, get

The result is If it is smaller than a certain constant, it contradicts setting L to be bounded, so it is obtained that L is bounded.

make

will be discussed later

I=1,...,n, introduce a new scaling transformation:

Similar to the scaling transformation mentioned above

Among them, it must be stated that:

b=[0,...,1],/>

A ^*T P ^* +P ^* A≤-2I

And the symmetric matrix P ^* is greater than 0, and the Lyapunov function is introduced for the above system

Substitute:

Simplified:

Integrate the right side of this expression

According to Barbalat Lemma

Step 4: Simulation verification results.

As shown in Figure 1, it is the actual x ₁ , x ₂ , and the observed value state response curve.

As shown in Figure 2, it is the actual x ₃ , x ₄ , and the observed value The state response curve of

As shown in Figure 3, it is the state response curve of L and u.

Claims (2)

1. The measuring output feedback control method of the fourth-order uncertain nonlinear mechanical arm system is characterized by comprising the following steps of:

step one: analyzing a mechanical arm system and establishing a corresponding system model;

the kinetic model of the robotic arm system is as follows:

the variable q as shown above ₁ Representing the displacement of the connecting rod in the system, q ₂ Representing the displacement of the rotor of the motor, J ₁ Representing the inertia of the connecting rod, J _m Representing the inertia, k, of the rotor of the motor ₀ Represents the elastic constant, g is the gravitational constant, m represents the mass, l ₀ Represents the mass center of the connecting rod, F ₁ Representing the viscous friction coefficient of the connecting rod, F _m Representing the viscous friction coefficient of the motor rotor, u being the torque transmitted by the motor, of which variables only q is present ₁ Is measurable;

the state space model of a four-stage mechanical arm system is as follows:

y＝θ(t)x ₁

there are some errors in the actual measurement, so the concept of sensitivity θ (t) is introduced here, and it is assumed that sensitivity θ (t) is continuous to 0+.θ ₁ ≤θ(t)≤θ ₂ Wherein θ is ₁ ，θ ₂ Is a known positive constant;

step two: the analysis system establishes a corresponding observer;

first, one identity matrix is I epsilon R ^4×4 To represent, then define matrices a, B, D,

θ (t) is the unknown measurement sensitivity, and the lower bound is the constant θ ₁ ，σ>0 is also a constant, h _i >0，k _i >0 is a design parameter;

lemma 1: for an arbitrary constant alpha>0 all have a constant h _i >0，V>0, and a digital matrix such that p=p ^T >0 is A ^T P+PA≤-αI,DP+PD≥VI；

And (4) lemma 2: for an arbitrary constant alpha>0 all have a constant k _i >0，β>0, and a digital matrix such that q=q ^T >0 is B ^T Q+QB≤-βI,DQ+QD>0；

Let i=1, 2,3 … n satisfy the linear growth condition: i f _i (t,v,x)|≤c(|x ₁ |+.......+|x _n I), wherein c>0, an unknown constant, called the unknown growth rate; based on the parameters obtained in quotients 1 and 2 and designing a dynamic outputThe feedback controller designs a standard form of a fourth-order system observer according to the mechanical arm system, wherein the standard form is as follows:

L(0)＝1

l is determined primarily by the above equation, wherein σ,is a constant of design and +.>0<σ<0.5,τ>1；

Step three: deducing that the system state is bounded and the output is finally kept in a preset range by using a dynamic scaling technology, a Lyapunov function and a linear matrix inequality method;

step four: and (5) simulating and verifying a result.

2. The method for feedback control of metrology output of a fourth order uncertainty nonlinear mechanical arm system in accordance with claim 1, wherein: the third step is specifically as follows:

when i=1, 2,3,4, letThen introducing a scaling transformation:

according to the scaling and transformingThe system is described as:

wherein ε= [ ε ] ₁ ,ε ₂ ,ε ₃ ,ε ₄ ] ^T ,H＝[h ₁ ,h ₂ ,h ₃ ,h ₄ ] ^T

f ₁ (t,v,x)＝0

f ₃ (t,v,x)＝0

For the assumed system, under the condition that the dynamic gain is bounded and the global convergence of other closed loop states is zero, the global self-adaptive state adjustment is realized through the control scheme formed by the above steps;

firstly, selecting a Lyapunov function:

from the following componentsDeducing:

when 0< σ <0.5, find the design parameter τ satisfies:

0.5α-m ₁ τ ^-2σ ≥γ ₁

βτ-kτ ^2σ -m ₂ ≥γ ₂

γ ₁ ,γ ₂ is a constant value, and is a function of the constant,

thus, it is possible to obtain:

γ ₁ L-c ₁ ,γ ₂ L-c ₂ possibly negative; therefore, the method is not a standard Lyapunov function, i.e. the asymptotic stability of the closed-loop system cannot be directly proved according to the Lyapunov stability theorem;

assuming that the solution has a maximum interval [0, t _f )，t _f ∈(0,+∞)；

Let L (t) be [0, t _f ) The inner limit will prove the progressive convergence of the progressive system;

proving that L is bounded:

let L be at [0, t _f ) The upper part 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 ₁ L-c ₁ ≥1，γ ₂ L-c ₂ ≥1

Combining the above to obtain

By passing throughDefinition of (1) to obtain

As a result, it was obtainedLess than a certain constant, contradicts the set L bounded, so that L is bounded;

order the

I=1, … …, n, introducing a new scaling transform:

A ^*T P ^* +P ^* A≤-2I

and symmetrical matrix P ^* Above 0, introducing a Lieplov function for the above system

Substituting to obtain:

simplifying and obtaining:

to the right integrate

According to the Barbalat lemma