CN112859600A - Mechanical system finite time control method based on extended state observer - Google Patents

Mechanical system finite time control method based on extended state observer Download PDF

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CN112859600A
CN112859600A CN202110025694.4A CN202110025694A CN112859600A CN 112859600 A CN112859600 A CN 112859600A CN 202110025694 A CN202110025694 A CN 202110025694A CN 112859600 A CN112859600 A CN 112859600A
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state observer
extended state
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sliding mode
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王立鹏
汤龙成
刘元基
杨超峰
杨世龙
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China Spaceflight Electronic Technology Research Institute
Aerospace Times Feihong Technology Co ltd
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Abstract

The invention belongs to the technical field of trajectory tracking, and particularly relates to a mechanical system finite time control method based on an extended state observer. The control method utilizes the nonlinear extended state observer to improve the robustness of the system, adopts the nonsingular terminal sliding controller to carry out fractional power reconstruction on sliding mode variables of the nonsingular terminal sliding controller, enables the system state to be rapidly converged to a balance point within limited time in a sliding stage, and realizes the control of a mechanical system. The method adopts the nonlinear extended state observer with the finite time convergence characteristic to improve the robustness of the system, adopts the nonsingular terminal sliding mode control method to realize the finite time control, and simultaneously performs fractional power reconstruction on sliding mode variables of the nonsingular terminal sliding mode to reduce buffeting, quickly estimates model uncertainty and external disturbance of the system and realizes the finite time convergence of the system state to the neighborhood of a balance point.

Description

Mechanical system finite time control method based on extended state observer
Technical Field
The invention belongs to the technical field of trajectory tracking, and particularly relates to a mechanical system finite time control method based on an extended state observer.
Technical Field
Uncertain second-order mechanical systems characterized by the Euler-Lagrange equation have many examples in the physical world, such as unmanned aerial vehicle systems, pneumatic servo systems, and rigid robotic arm systems. The finite time trajectory tracking control method has been widely studied in recent years because it has a feature of promoting the system state to quickly track a desired trajectory. In order to reduce the influence of uncertainty and external disturbance of a system model on control precision, the invention adopts a nonlinear extended state observer with limited time convergence characteristics to improve the robustness of the system.
The nonsingular terminal sliding mode control method is a core technology for realizing the purpose of finite time control. The sliding mode control is divided into an approach stage and a sliding stage, and in order to solve the problem of gradual convergence of the system state in the sliding stage in the traditional sliding mode control, fractional power reconstruction is carried out on sliding mode variables of a nonsingular terminal sliding mode, so that the system state can be quickly converged to a balance point within limited time in the sliding stage. In a traditional nonsingular terminal sliding mode controller, variable structure control is adopted as a robust item to restrain uncertainty and external disturbance of a system model. However, variable configuration control may excite high frequency buffeting of the controller, damage the actuator, and even cause system divergence.
Disclosure of Invention
The invention discloses a mechanical system finite time control method based on an extended state observer, which aims to solve any of the above problems or other potential problems in the prior art.
In order to solve the problems, the technical scheme of the invention is as follows: a mechanical system finite time control method based on an extended state observer specifically comprises the following steps:
s1) collecting position signal and speed signal of sensor device measuring system, comparing with target expected track value, and obtaining position signal error valueδ1And speed signal error value delta2；
Then the position signal and the estimated signal z of the extended state observer are compared1Comparing to obtain the estimation error e of the extended state observer1；
S2) will S1) get the position signal error value δ1And speed signal error value delta2As a sliding mode control quantity model, the error e of the obtained extended state observer is used1As the input of the extended state observer and the input of the extended state observer, the output s of the sliding mode control quantity and the estimated signal value z of the extended state observer are respectively obtained3；
S3) obtaining the output S of the sliding mode control quantity obtained in S2) and the estimated signal value z of the extended state observer3The output signal of the finite time controller is obtained as the input of the nonsingular terminal sliding mode controller;
s4) sends the output signal as a control signal to an actuator of the mechanical system.
Further, the step S1) includes the following steps:
s1.1) acquiring position signals x of a sensor device measurement system1Sum velocity signal x2，
S1.2) establishing an expected trajectory value xd,
S1.3) obtaining the position signal x1Sum velocity signal x2And establishing a desired trajectory value xdDifference is made to obtain a position signal error value delta1And speed signal error value delta2，
S1.4) and then the position signal x1And the estimated signal z of the extended state observer1Comparing to obtain the estimation error e of the extended state observer1。
Further, in S1.3), a position signal error value δ is obtained by equation (8)1And speed signal error value delta2The formula is as follows:
Figure BDA0002890168990000031
in the formula, xdTo the desired trajectory value, z2The signal value is estimated for the velocity of the extended state observer.
Further, the estimation error e of the extended state observer in S1.4)1The calculation is performed by the following equation (6):
e1(t)＝z1(t)-x1(t) (6)，
in the formula, z1(t) is a position estimation signal of the extended state observer, t being time.
Further, z is1(t) is obtained by the following formula, as shown below,
Figure BDA0002890168990000032
in the formula (I), the compound is shown in the specification,
Figure BDA0002890168990000033
is the real-time output quantity of the extended state observer, and when the time t is equal to 0, the initial value z is1(0)＝0,z1(t)。
Further, the specific steps of S2) are:
s2.1) obtaining a position signal error value delta from S1.3)1And speed signal error value delta2As the input of the sliding mode control quantity model, the output s of the sliding mode control quantity is obtained,
s2.2) obtaining the estimation error e of the extended state observer by the S1.4)1As an input of the extended state observer, an estimated signal value z of the extended state observer is obtained3。
Further, the control model in S2.1) is expressed by formula (10), which is as follows:
Figure BDA0002890168990000034
in the formula, s is the output of the sliding mode control quantity, sigma is a parameter, and sigma is>0, γ ═ p/q, p and q are positive odd variables, p ═ 2m +1, and p is>q, m are largeA positive integer at 0; sigγ(. cndot.) is a fractional power sign function.
Further, the extended state observer in said S2.2) is expressed by the formula (5), which is shown below,
Figure BDA0002890168990000041
in the formula, betaiTo extend the gain of the state observer, i ═ 1.2.3, e1To extend the state observer error signal, z1For the position estimation signal of the extended state observer, z2For the velocity estimation signal of the extended state observer, M0Is a known nominal part of the inertial matrix, which is a dynamic model of the mechanical system, and τ is a finite time controlled quantity.
Further, the specific steps of S3) are:
s3.1) obtaining the sliding mode control quantity output S from S2.1), and obtaining the estimated signal z of the extended state observer from S2.2)3And S1.3) position signal error value delta1And speed signal error value delta2As input of the nonsingular terminal sliding mode controller;
and S3.2) outputting a control signal by the nonsingular terminal sliding mode controller.
Further, the nonsingular terminal sliding mode controller in S3.1) obtains the finite time control quantity τ by the following formula (16), which is as follows:
Figure BDA0002890168990000042
in the formula (I), the compound is shown in the specification,
Figure BDA0002890168990000043
is a robust term of the system, K1And K2Is the normal constant gain.
The invention has the beneficial effects that: due to the adoption of the technical scheme, the method adopts the nonlinear extended state observer with the finite time convergence characteristic to improve the robustness of the system, adopts the nonsingular terminal sliding mode control method to achieve the finite time control purpose, and simultaneously performs fractional power reconstruction on sliding mode variables of the nonsingular terminal sliding mode to reduce buffeting, quickly estimates model uncertainty and external disturbance of the system and achieves that the finite time of the system state converges into the neighborhood of a balance point.
Drawings
Fig. 1 is a schematic flow chart of a finite time control method of an uncertain second-order mechanical system based on an extended state observer according to the present invention.
Fig. 2 is a performance effect diagram of the extended state observer provided by the present invention.
Fig. 3 is a diagram illustrating the tracking effect of the first joint position of the robot arm provided by the present invention.
Fig. 4 is a diagram illustrating the tracking effect of the second joint position of the robot arm provided by the present invention.
Detailed Description
In order to make the technical problems, technical solutions and advantages to be solved by the present invention clearer, the following detailed description is made with reference to specific steps, simulations and drawings.
Step S1, using the sensor device to measure the position signal and the speed signal of the system, the expression of the position signal and the speed signal is as follows:
the dynamic model of the second order uncertain system was established as follows:
Figure BDA0002890168990000051
in the formula (I), the compound is shown in the specification,
Figure BDA0002890168990000052
and
Figure BDA0002890168990000053
representing generalized position coordinates, velocity, and acceleration;
Figure BDA0002890168990000054
is an inertia matrix, where the nominal inertia matrix M0(q) is symmetrically positive, MΔ(q) represents the unknown part and,
Figure BDA0002890168990000055
which represents the term of the centrifugal coriolis force,
Figure BDA0002890168990000056
the term of the force of gravity is represented,
Figure BDA0002890168990000057
representing an unknown and bounded external disturbance,
Figure BDA0002890168990000061
representing the joint control input torque.
The kinetic model equation (1) can be rewritten as:
Figure BDA0002890168990000062
in the formula (I), the compound is shown in the specification,
Figure BDA0002890168990000063
is a composite disturbance of the mechanical system consisting of external disturbances and parameter uncertainty.
Define x separately1(t) q (t) and
Figure BDA0002890168990000064
then equation (2) can be rewritten as:
Figure BDA0002890168990000065
step S2, calculating the estimation signal of the extended state observer to the uncertainty and external non-disturbance of the system model, the process is as follows:
estimating the complex disturbance l of the mechanical system by using the extended state observerlDefined as a new state x3Then the system (3) can be rewritten as
Figure BDA0002890168990000066
For the system (4), the invention envisages the following extended state observer
Figure BDA0002890168990000067
Wherein
Figure BDA0002890168990000068
And 0<α<1 is an adjustable parameter, zi(i-1, 2,3) is used to estimate the state xi,βiIs the gain of the observer.
The observer error formula is defined as follows:
e1(t)＝z1(t)-x1(t) (6)
in the formula, z1(t) is a position estimation signal of the extended state observer, and when the time t is 0, the initial value z is set to be1(0)＝0,z1(t) is updated by the formula
Figure BDA0002890168990000069
Figure BDA00028901689900000610
Is the real-time output of the extended state observer (5).
Obtained from (4) and (5), the error system of the observer is
Figure BDA0002890168990000071
Step S3, calculating the nonsingular terminal sliding mode signal by using the system state, wherein the process is as follows:
based on the observed quantity z2And z3The invention designs a controller based on a nonsingular terminal sliding mode surface, so that the system state can reach a tracking target in a certain time.
Defining the tracking error of the system as
Figure BDA0002890168990000072
Wherein xdIs the desired trajectory.
The derivation of (8) and the substitution of (4) are carried out, the error equation of the system is
Figure BDA0002890168990000073
Designing nonsingular terminal sliding mode variables as
Figure BDA0002890168990000074
Where the parameter σ >0, γ ═ p/q, p and q are positive odd variables, p ═ 2m +1 is satisfied, and p > q.
Step S4, calculating an output signal of the finite time controller:
the driving system (4) of the nonsingular terminal sliding mode controller based on the extended state observer can be designed to reach the sliding mode surface by the aid of the driving system (10).
The time derivative of the sliding mode variable (10) is
Figure BDA0002890168990000075
Bringing (9) into (11), rewritable
Figure BDA0002890168990000081
When in use
Figure BDA0002890168990000082
The kinetic equation of the system is equivalent to a non-linear equation
Figure BDA0002890168990000083
According to the design method of the sliding mode controller, the control moment tau is taueq+τsmc. If the complex disturbance l of the system is precisely known, then
Figure BDA0002890168990000084
Can calculate taueq：
Figure BDA0002890168990000085
But l is in practice uncertain, so based on the extended state observer and boundary layer techniques the following controllers can be designed:
Figure BDA0002890168990000086
thus, in practice the controller may be designed to
Figure BDA0002890168990000087
Wherein z is3Is the output of the extended state observer, used to estimate the complex disturbance l,
Figure BDA0002890168990000088
is a robust term of the system, K1And K2Is the normal constant gain.
To better demonstrate that the controller may be a system with limited time convergence, system stability verification is described in detail below. The device comprises two parts: a stability certification of the extended state observer and a stability certification of the finite time controller. Wherein, the state convergence time of the extended state observer should be theoretically shorter than the state convergence time of the system.
Introduction 1: for any x and non-zero real y, the following inequality holds,
0≤|x|(1-tanh(|x/y|))≤a*|y| (17)
wherein, a*＝0.2785。
2, leading: for the system
Figure BDA0002890168990000091
Wherein
Figure BDA0002890168990000092
If a positive real scalar function V (x) is defined, such that
Figure BDA0002890168990000093
The system is stable for a limited time. And convergence time is
Figure BDA0002890168990000094
Theorem 1: for the extended state observer (5), there is an adjustable parameter β1,β2,β3And α, in the presence of
Figure BDA0002890168990000095
In this case, the error of the observer may converge into the neighborhood of zero in a finite time.
And (3) proving that:
selecting Lyapunov functions
Figure BDA0002890168990000096
Wherein
Figure BDA0002890168990000097
Represents a vector eiThe j element, P being
Figure BDA0002890168990000098
ηjTime ofThe derivative is
Figure BDA0002890168990000099
Wherein
Figure BDA00028901689900000910
B＝[0,0,1]T,
Figure BDA00028901689900000911
The feature matrix of A is
Figure BDA0002890168990000101
When beta is1,β2,β3And alpha satisfies
Figure BDA0002890168990000102
When the temperature of the water is higher than the set temperature,
Figure BDA0002890168990000103
if positive, A is a Hurwitz matrix, and the eigenvalues of A have positive real parts, P is a positive matrix.
To V1Is derived by
Figure BDA0002890168990000104
Wherein
Figure BDA0002890168990000105
In fact, having a root with a negative real part is equivalent to having a positive definite matrix Q, such that P is a solution of the lyapunov equation.
ATP+PA＝-Q (22)
According to (19), the following inequality holds
Figure BDA0002890168990000106
According to (21) and (22), the
Figure BDA0002890168990000111
If the condition is
Figure BDA0002890168990000112
Satisfy, then
Figure BDA0002890168990000113
When in use
Figure BDA0002890168990000114
And 0<θ0<1 is, then
Figure BDA0002890168990000115
Wherein
Figure BDA0002890168990000116
q2,min＝min{q2j,j＝1,2,…,n}。
According to (23) and inequality
Figure BDA0002890168990000117
Then
Figure BDA0002890168990000118
This is true. Therefore, the temperature of the molten metal is controlled,
Figure BDA0002890168990000119
error η of observer according to lemma 2jWill converge in a finite time, the time expression is
Figure BDA00028901689900001110
Step 6: and (3) carrying out stability analysis on the finite time controller, wherein the process is as follows:
theorem 2: for the system (9), a nonsingular terminal sliding mode controller (16) and a sliding mode variable (10) are designed, and a proper parameter K is selectede,K1Satisfy the requirement of
Figure BDA00028901689900001111
And K ise<K1The tracking error of the closed loop system can converge to the neighborhood of zero in a finite time.
And (3) proving that:
the (16) is brought into (9), so that a closed-loop system can be obtained
Figure BDA0002890168990000121
Selecting Lyapunov functions
Figure BDA0002890168990000122
Bringing the controller (16) into (12) can obtain
Figure BDA0002890168990000123
To V2Derivative, bring (30) into
Figure BDA0002890168990000124
According to theorem 1, z is known3Is bounded and there is a suitable parameter kej(j ═ 1, 2.. times, n) satisfies
Figure BDA0002890168990000125
Is obtained by unfolding (31)
Figure BDA0002890168990000126
Wherein
Figure BDA0002890168990000127
k3,min＝min{k3j,j＝1,2,…,n}，
Figure BDA0002890168990000128
If it is not
Figure BDA0002890168990000129
Then epsilon is less than or equal to 2 theta2λmin(K3)V2. Then (32) can be rewritten as
Figure BDA00028901689900001210
According to lemma 2, the system state may be at a finite time tf2Internally converging to the vicinity of the equilibrium point, and the time expression is
Figure BDA0002890168990000131
Example (b):
in order to verify the effectiveness of the proposed method, the invention gives a specific simulation process as follows:
the double-link rigid mechanical arm is adopted in the simulation to verify the effectiveness of the proposed controller, and the structural block diagram of the system is shown in fig. 1. Definition of x1＝[x11,x12]TAs the joint angle of the mechanical arm, then the correlation matrix in the two-link robot mathematical model is given:
Figure BDA0002890168990000132
Figure BDA0002890168990000133
Figure BDA0002890168990000134
in the formula
Figure BDA0002890168990000135
p3＝m2l1lc2，p4＝m1lc2+m2l1，p5＝m2lc2；miAnd liMass and length of the connecting rod i, m1＝2.00(kg),m2＝0.85(kg)，l1＝0.35(m),l2＝0.31(m)；IiIs the moment of inertia of the connecting rod i,
Figure BDA0002890168990000136
lciis the centroid of the ith link; g is 9.8 (m/s)2)。
The initial position and speed of the robot are:
x11(0)＝x12(0)＝1(rad)，x21(0)＝x22(0)＝0(rad/s)
the set desired trajectory is:
Figure BDA0002890168990000137
the disturbance torque is:
d(t)＝[0.1sin(0.5t)+0.25cos(0.5t),0.25sin(0.5t)+0.1sin(0.5t)]T
the parameters of the extended state observer are: α is 0.5, β1＝300,β2＝10,β325; the controller has parameters σ 50, p 9, q 7, and K1＝diag{150,150},K2＝diag{150,150},ρ＝0.1。
As shown in fig. 2, compared with the conventional state observer, the extended state observer can converge the system error to the neighborhood of zero in milliseconds, and the convergence time is much shorter than that of the system state; as shown in FIGS. 3 and 4, the position state error of the system can be converged to the zero neighborhood within a limited time, and the convergence precision reaches 10-3A rank.
The foregoing description shows and describes several preferred embodiments of the present application, but as aforementioned, it is to be understood that the application is not limited to the forms disclosed herein, but is not to be construed as excluding other embodiments and is capable of use in various other combinations, modifications, and environments and is capable of changes within the scope of the application as described herein, commensurate with the above teachings, or the skill or knowledge of the relevant art. And that modifications and variations may be effected by those skilled in the art without departing from the spirit and scope of the application, which is to be protected by the claims appended hereto.

Claims (10)

1. A mechanical system finite time control method based on an extended state observer is characterized by specifically comprising the following steps:
s1) collecting position signals and speed signals of a measuring system of the sensor equipment, comparing the position signals and the speed signals with a target expected track value to obtain position signal error values delta1And speed signal error value delta2；
Then the position signal and the estimated signal Z of the extended state observer are compared1Comparing to obtain the estimation error e of the extended state observer1；
S2) will S1) get the position signal error value δ1And speed signal error value delta2As a sliding mode control quantity model, the error e of the obtained extended state observer is used1As the input of the extended state observer and the input of the extended state observer, the output s of the sliding mode control quantity and the estimated signal value z of the extended state observer are respectively obtained3；
S3) obtaining the output S of the sliding mode control quantity obtained in S2) and the estimated signal value z of the extended state observer3The output signal of the finite time controller is obtained as the input of the nonsingular terminal sliding mode controller;
s4) sends the output signal as a control signal to an actuator of the mechanical system.
2. The control method according to claim 1, wherein the step S1) includes the steps of:
s1.1) acquiring position signals x of a sensor device measurement system1Sum velocity signal x2，
S1.2) establishing an expected trajectory value xd，
S1.3) obtaining the position signal x1Sum velocity signal x2And establishing a desired trajectory value xdDifference is made to obtain a position signal error value delta1And speed signal error value delta2，
S1.4) and then the position signal x1And the estimated signal z of the extended state observer1Comparing to obtain the estimation error e of the extended state observer1。
3. Method according to claim 2, characterized in that in S1.3) a position signal error value δ is determined by means of equation (8)1And speed signal error value delta2The formula is as follows:
Figure FDA0002890168980000021
in the formula, xdTo the desired trajectory value, z2The signal value is estimated for the velocity of the extended state observer.
4. Method according to claim 2, characterized in that the estimation error e of the extended state observer in S1.4) is1The calculation is performed by the following equation (6):
e1(t)＝z1(t)-x1(t) (6)，
in the formula, z1(t) is expansionAnd (c) a position estimation signal of a state observer, wherein t is time.
5. The method of claim 4, wherein z is1(t) is obtained by the following formula, as shown below,
Figure FDA0002890168980000022
in the formula (I), the compound is shown in the specification,
Figure FDA0002890168980000023
is the real-time output quantity of the extended state observer, and when the time t is equal to 0, the initial value z is1(0)＝0，z1(t)。
6. The control method according to claim 2, characterized in that the specific steps of S2) are:
s2.1) obtaining a position signal error value delta from S1.3)1And speed signal error value delta2As the input of the sliding mode control quantity model, the output s of the sliding mode control quantity is obtained,
s2.2) obtaining the estimation error e of the extended state observer by the S1.4)1As an input of the extended state observer, an estimated signal value z of the extended state observer is obtained3。
7. The control method according to claim 6, characterized in that the control model in S2.1) is expressed by formula (10) as follows:
Figure FDA0002890168980000031
in the formula, s is a sliding mode control quantity output, sigma is a parameter, sigma is more than 0, gamma is p/q, p and q are positive odd variables, p is 2m +1, p is more than q, and m is a positive integer which is more than 0; sigγ(. cndot.) is a fractional power sign function.
8. Control method according to claim 6, characterized in that the extended state observer in S2.2) is expressed by the formula (5), which is shown below,
Figure FDA0002890168980000032
in the formula, betaiTo extend the gain of the state observer, i ═ 1.2.3, e1To expand the state observer error signal, Z1For the position estimation signal of the extended state observer, Z2For the velocity estimation signal of the extended state observer, M0Is a known nominal part of the inertial matrix, which is a dynamic model of the mechanical system, and τ is a finite time controlled quantity.
9. The control method according to claim 1, wherein the specific steps of S3) are:
s3.1) obtaining the sliding mode control quantity output S from S2.1), and obtaining the estimated signal z of the extended state observer from S2.2)3And S1.3) position signal error value delta1And speed signal error value delta2As input of the nonsingular terminal sliding mode controller;
and S3.2) outputting a control signal by the nonsingular terminal sliding mode controller.
10. The control method according to claim 9, characterized in that the nonsingular terminal sliding mode controller in S3.1) finds the finite time control quantity τ by the following equation (16) as follows:
Figure FDA0002890168980000041
in the formula (I), the compound is shown in the specification,
Figure FDA0002890168980000042
is a robust term of the system, K1And K2Is the normal constant gain.
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