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:
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,
in the formula (I), the compound is shown in the specification,
is the real-time output quantity of the extended state observer, and when the time t is equal to 0, the initial value z is
1(0)=0,z
1(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:
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,
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:
in the formula (I), the compound is shown in the specification,
is a robust term of the system, K
1And K
2Is 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.
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:
in the formula (I), the compound is shown in the specification,
and
representing generalized position coordinates, velocity, and acceleration;
is an inertia matrix, where the nominal inertia matrix M
0(q) is symmetrically positive, M
Δ(q) represents the unknown part and,
which represents the term of the centrifugal coriolis force,
the term of the force of gravity is represented,
representing an unknown and bounded external disturbance,
representing the joint control input torque.
The kinetic model equation (1) can be rewritten as:
in the formula (I), the compound is shown in the specification,
is a composite disturbance of the mechanical system consisting of external disturbances and parameter uncertainty.
Define x separately
1(t) q (t) and
then equation (2) can be rewritten as:
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
For the system (4), the invention envisages the following extended state observer
Wherein
And 0<α<1 is an adjustable parameter, z
i(i-1, 2,3) is used to estimate the state x
i,β
iIs the gain of the observer.
The observer error formula is defined as follows:
e1(t)=z1(t)-x1(t) (6)
in the formula, z
1(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 be
1(0)=0,z
1(t) is updated by the formula
Is the real-time output of the extended state observer (5).
Obtained from (4) and (5), the error system of the observer is
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
Wherein xdIs the desired trajectory.
The derivation of (8) and the substitution of (4) are carried out, the error equation of the system is
Designing nonsingular terminal sliding mode variables as
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
Bringing (9) into (11), rewritable
When in use
The kinetic equation of the system is equivalent to a non-linear equation
According to the design method of the sliding mode controller, the control moment tau is tau
eq+τ
smc. If the complex disturbance l of the system is precisely known, then
Can calculate tau
eq:
But l is in practice uncertain, so based on the extended state observer and boundary layer techniques the following controllers can be designed:
thus, in practice the controller may be designed to
Wherein z is
3Is the output of the extended state observer, used to estimate the complex disturbance l,
is a robust term of the system, K
1And K
2Is 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
Wherein
If a positive real scalar function V (x) is defined, such that
The system is stable for a limited time. And convergence time is
Theorem 1: for the extended state observer (5), there is an adjustable parameter β1,β2,β3And α, in the presence of
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
Wherein
Represents a vector e
iThe j element, P being
ηjTime ofThe derivative is
The feature matrix of A is
When beta is1,β2,β3And alpha satisfies
When the temperature of the water is higher than the set temperature,
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
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
According to (21) and (22), the
If the condition is
Satisfy, then
When in use
And 0<θ
0<1 is, then
Wherein
q
2,min=min{q
2j,j=1,2,…,n}。
According to (23) and inequality
Then
This is true. Therefore, the temperature of the molten metal is controlled,
error η of observer according to lemma 2jWill converge in a finite time, the time expression is
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 selected
e,K
1Satisfy the requirement of
And K is
e<K
1The 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
Selecting Lyapunov functions
Bringing the controller (16) into (12) can obtain
To V2Derivative, bring (30) into
According to
theorem 1, z is known
3Is bounded and there is a suitable parameter k
ej(j ═ 1, 2.. times, n) satisfies
Is obtained by unfolding (31)
Wherein
k
3,min=min{k
3j,j=1,2,…,n},
If it is not
Then epsilon is less than or equal to 2 theta
2λ
min(K
3)V
2. Then (32) can be rewritten as
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
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:
in the formula
p
3=m
2l
1l
c2,p
4=m
1l
c2+m
2l
1,p
5=m
2l
c2;m
iAnd l
iMass and length of the connecting rod i, m
1=2.00(kg),m
2=0.85(kg),l
1=0.35(m),l
2=0.31(m);I
iIs the moment of inertia of the connecting rod i,
l
ciis 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:
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.