CN114047694B - Self-adaptive output feedback control method for single-link mechanical arm - Google Patents
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Abstract
The invention discloses a self-adaptive output feedback control method for a single-link mechanical arm. The method comprises the following steps: firstly, modeling a single-connecting-rod mechanical arm to obtain a state space model of the mechanical arm; then constructing a corresponding state compensator aiming at the state of the system which cannot be measured, introducing a self-adaptive dynamic high gain, and estimating the state of the system; then, carrying out coordinate transformation on the system, and introducing an adjustable constant gain to obtain a new transformed system; then, an adaptive output feedback controller is designed for the new system based on a double control method; and finally, determining the value range of the adjustable constant gain by utilizing the Lyapunov stability theory, and realizing a global calm closed loop system. According to the invention, under the condition that the single-link mechanical arm has uncertain parameters, the unknown state of the system can be well estimated based on the self-adaptive thought, and the nonlinear mechanical arm system can be effectively controlled, so that the nonlinear mechanical arm system can tend to be stable under the influence of random disturbance.
Description
Technical Field
The invention belongs to the technical field of robots, and particularly relates to a self-adaptive output feedback control method for a single-link mechanical arm.
Background
Along with the continuous development of science and technology to the intelligent direction, the robot has become a high and new technology industry integrating multiple subjects such as computer, information and sensing technology, control theory, artificial intelligence and the like, and plays a great role in improving the industrial automation level. A robot arm, which is a typical representative of an industrial robot, can simulate some motion functions of a human hand and an arm, and is an automatic operation device for gripping, carrying an object or operating a tool according to a fixed program. The device can replace heavy labor of people to realize mechanization and automation of production, and can be operated under harmful environment to ensure personal safety, thereby being widely applied to departments of mechanical manufacture, metallurgy, electronics, light industry and the like.
Single link robotic arms can be modeled as a class of nonlinear systems. Because the time lag phenomenon of the system is often caused by the measurement of the sensor and the delay of signal transmission, which can greatly affect the performance of the system, how to design a controller capable of counteracting the time lag effect of the system for a single-link mechanical arm is more and more important. In the analysis of nonlinear time-lapse systems, the most commonly used tool is the Lyapunov-Krasovskii functional, by means of which a number of analytical methods of nonlinear time-lapse systems are generated. Currently, the upper-bound known studies on derivatives of time-varying lags are quite mature. However, when upper bound information is not available, related studies are still not involved, and a solution is needed.
In practical application, a single-link mechanical arm can be further modeled into a random uncertain nonlinear time-lapse system due to a plurality of random uncertainty factors such as environmental factors, climate factors, mechanical faults, component aging, electronic noise and the like, and a plurality of students at home and abroad begin to study related control methods. Uncertainty factors are objectively present and difficult to avoid, and can have a significant impact on the system. Adaptive control is one of the effective methods of studying uncertain nonlinear systems. The adaptive controller can modify its own characteristics to accommodate changes in the dynamic characteristics of the object and disturbance, solving the "uncertainty" in the system by a time-varying dynamic gain.
On the other hand, because of the limitations of metrology technology and metrology tools, sensors cannot accurately detect the state of the system, and accurate relationships between the true output of the system and the state are difficult to obtain. Thus, the functional relationship between the system metrology output and the system state for designing a controller is often unknown, and the unknowns of such output functions present a significant challenge to controller design. Some of the studies that exist rely on system output functions that may be slightly or locally Lipschitz continuous everywhere, but are difficult to meet in many practical systems.
Disclosure of Invention
The invention aims to provide a self-adaptive output feedback control method for a single-link mechanical arm, which can estimate the unknown state of a system, and realize effective control of a nonlinear mechanical arm system so that the nonlinear mechanical arm system can tend to be stable under the influence of random disturbance.
The technical solution for realizing the purpose of the invention is as follows: an adaptive output feedback control method for a single-link mechanical arm is characterized by comprising the following steps:
step 1, modeling a single-link mechanical arm to obtain an uncertain nonlinear system model with random disturbance influence, wherein uncertain factors comprise an unknown growth rate, an unknown control direction, an unknown output and an unknown time lag;
step 2, aiming at the system state which cannot be measured, designing a dynamic compensator to estimate the system state;
step 3, designing a self-adaptive output feedback controller based on the estimated state and system output;
and 4, determining the range of the adjustable constant gain M introduced in the coordinate transformation by utilizing the Lyapunov stability theory, and controlling the single-link mechanical arm.
Compared with the prior art, the invention has the remarkable advantages that: (1) Modeling a single-connecting-rod mechanical arm to obtain a model of a random nonlinear time-lapse system, and designing a dynamic compensator and a self-adaptive output feedback controller aiming at the model; (2) The limiting condition applied to the unknown output function is greatly relaxed, namely the output function is not required to be tiny or Lipschitz continuous; (3) There is no need to know the derivative upper bound information of the growth rate and time-varying time lag imposed on the unknown nonlinear function; (4) The value range of the parameter is determined by utilizing the Lyapunov stability theory, the unknown state of the system can be well estimated, and the closed loop system is globally calmed.
Drawings
Fig. 1 is a flow chart of the adaptive output feedback control method for the single-link mechanical arm.
Fig. 2 is a schematic diagram of a system structure of a single link mechanical arm model in the present invention.
Fig. 3 is a schematic diagram of a system structure of a single link mechanical arm in the present invention.
Fig. 4 is a graph of the trajectory of the system output y (t) in the present invention.
Fig. 5 is a graph of the trace of the adaptive dynamic gain L (t) in the present invention.
FIG. 6 is a system compensator error e in the present invention 1 (t),e 2 (t),e 3 A graph of the variation of (t).
FIG. 7 is a system state x in the present invention 1 (t),x 2 (t),x 3 A graph of the variation of (t).
Fig. 8 is a graph of the variation of the system control input u (t) in the present invention.
Detailed Description
The invention discloses a self-adaptive output feedback control method for a single-link mechanical arm, which comprises the following steps of:
step 1, modeling a single-link mechanical arm to obtain an uncertain nonlinear system model with random disturbance influence, wherein uncertain factors comprise an unknown growth rate, an unknown control direction, an unknown output and an unknown time lag;
step 2, aiming at the system state which cannot be measured, designing a dynamic compensator to estimate the system state;
step 3, designing a self-adaptive output feedback controller based on the estimated state and system output;
and 4, determining the range of the adjustable constant gain M introduced in the coordinate transformation by utilizing the Lyapunov stability theory, and controlling the single-link mechanical arm.
Further, the system model established in step 1 is specifically as follows:
step 1.1, the system dynamics of a single-link mechanical arm are as follows:
wherein q is,Respectively representing the position, the speed and the acceleration of the connecting rod; τ r Torque generated for the electrical system; u is a control input for representing an electromechanical torque; />Random disturbance ω with time-varying delay d (t) and torque defined in the system; d is mechanical inertia; b is the viscous friction coefficient of the joint; n is a positive constant related to load mass and gravity coefficient; m is electricityA pivot inductance; h is the armature resistance; k (K) m Is the back electromotive force coefficient;
order theModeling a single-link mechanical arm as:
further convert into:
dx 1 ＝x 2 dt
y＝h(x 1 )
the above system is represented as a special class of systems, namely random nonlinear time-lapse systems with unknown output functions:
y＝h(x 1 )
wherein x is 1 ,x 2 And x 3 In order to be in the state of the system,u and y are the input and output of the system, respectively; unknown function phi i (. Cndot.) and->A continuous nonlinear function with an initial value of 0; h (·) is the output function of the system; d (t) is a time lag link of the system; omega is probability space +.>Standard wiener process;
step 1.2, for the system model established in step 1.1, assume the following:
assuming 1, the presence of unknown positive constants d and τ causes the time-varying time-lag d (t) to satisfy 0d (t) < d and
suppose 2, for an unknown nonlinear function in the system i (. Cndot.) A. Cndot. CPresence of unknown positive constant θ 1 And theta 2 So that the following formula holds:
wherein ii represents the norm;
let 3, the continuous output function hs) satisfy h (0) =0, and a constant ρ exists 1 Not less than 0 and ρ 2 More than or equal to 0 so that the absolute value of the absolute value is less than or equal to rho 1 s+ρ 2 。
Further, the dynamic compensator in step 2 is specifically designed as follows:
step 2.1, designing a dynamic compensator based on the system model established in the step 1.1, wherein the dynamic compensator comprises the following steps:
in the method, in the process of the invention,for compensator state, L is dynamic high gain and L (0) =1, parameter a i > 0, i=1, 2,3, the presence of l > 0 being such that a T P+PA is less than or equal to- (l+2) I, wherein P is positive definite matrix, and ∈A>
2.2, carrying out the following coordinate transformation on the system model established in the step 1.1:
in e i I=1, 2,3, representing an estimation error; z i I=1, 2,3, representing a state after the system coordinate transformation; v represents input after system coordinate transformation, M is more than or equal to 1, adjustable normal gain is achieved, and sigma is more than or equal to 0 and is any constant;
the system transforms into the following form:
wherein the method comprises the steps of
e＝[e 1 ,e 2 ,e 3 ] T ,Z＝[Z 1 ,Z 2 ,z 3 ] T ,B 1 ＝[a 1 ,a 2 ,a 3 ] T ,B 2 ＝[0,0,1] T ,B 3 ＝[1,0,0] T ,
Further, the design adaptive output feedback controller described in step 3 is in the form of:
v＝-L -σ b 3 y-b 2 z 2 -b 1 z 3
wherein b is i > 0, i=1, 2,3, for the system control coefficients such thatWherein P is z Is positive matrix, ++>
Further, the selection method of the adjustable constant gain M in step 4 is as follows:
firstly, selecting a Lyapunov function, then selecting an adjustable constant gain M on the basis of a dynamic compensator and an adaptive output feedback controller designed in the step 2 and the step 3 by utilizing the Lyapunov stability theory to ensure that the system is stable, namely, for the designed Lyapunov function V, the Earthway differential thereof meets the following conditionsWherein c 1 And c 2 Is a positive constant.
The invention will be described in further detail with reference to the accompanying drawings and specific examples.
Examples
Referring to fig. 1, the adaptive output feedback control method for a single-link mechanical arm of the present invention includes the following steps:
step 1, a single-link mechanical arm model is established to obtain a random nonlinear time-lapse system model, and the method is combined with fig. 2-3, and specifically comprises the following steps:
step 1.1, the system dynamics of a single-link mechanical arm are as follows:
wherein q is,Respectively representing the position, the speed and the acceleration of the connecting rod; τ r Torque generated for the electrical system;random disturbance ω with time-varying delay d (t) and torque defined in the system; u is a control input for representing an electromechanical torque; d is mechanical inertia; b is the viscous friction coefficient of the joint; n is a positive constant related to load mass and gravity coefficient; m is armature inductance; h is the armature resistance; k (K) m Is the back emf coefficient.
Order theThe single link robotic arm can be modeled as:
further can be converted into:
by the above-mentioned transition, it can be derived that the system (1) with an unknown control direction is equivalent to the system (2) with an unknown output function. The above system (2) can be expressed as a special class of systems, namely a random nonlinear time-lapse system with an unknown output function:
y＝h(x 1 )
wherein x is 1 ,x 2 And x 3 In order to be in the state of the system,u and y are the input to the system and the output of the system, respectively; phi (phi) 1 ＝0，/> h (·) is the output function of the system; d (t) is a time lag link of the system.
The design targets of the controller are as follows: the adaptive output feedback controller is designed such that the system is almost globally bounded:
v＝-L -σ b 3 y-b 2 z 2 -b 1 z 3
wherein b is i > 0, i=1, 2,3, for the system control coefficients such thatWherein P is z Is positive matrix, ++>
Step 1.2, the following assumption is made on the system, and the design flow of the controller is simplified:
assuming 1, the presence of unknown positive constants d and τ causes the time-varying time-lag d (t) to satisfy 0d (t) < d and
suppose 2, for an unknown nonlinear function in the system i (. Cndot.) A. Cndot. CPresence of unknown positive constant θ 1 And theta 2 So that the following equation holds:
where II represents the norm.
Let 3, the continuously unknown output function hs) satisfy h (0) =0, and a constant ρ exists 1 Not less than 0 and ρ 2 More than or equal to 0 so that the absolute value of the absolute value is less than or equal to rho 1 s+ρ 2 。
The dynamic compensator in the step 2 is specifically designed as follows:
step 2.1, designing a dynamic compensator for estimating the system state which is not measurable, wherein the method comprises the following steps:
in the method, in the process of the invention,for compensator state, L is a dynamically adjustable gain and L (0) =1, parameter a i > 0, i=1, 2,3, the presence of l > 0 being such that a T P+PA is less than or equal to- (l+2) I, wherein P is positive definite matrix, and ∈A>
2.2, carrying out the following coordinate transformation on the system model established in the step 1.1:
in e i I=1, 2,3, representing the estimation error, z i I=1, 2,3, and v represent the state and input after the system coordinate transformation, respectively, m+.1 represents an adjustable normal gain, σ+.0 is an arbitrary constant.
The system transforms into the following form:
wherein the method comprises the steps of
e＝[e 1 ,e 2 ,e 3 ] T ,z＝[z 1 ,z 2 ,z 3 ] T ,B 1 ＝[a 1 ,a 2 ,a 3 ] T ,B 2 ＝[0,0,1] T ,B 3 ＝[1,0,0] T ,
Step 3, designing a self-adaptive output feedback controller based on the estimated state and system output, wherein the self-adaptive output feedback controller is specifically as follows:
v＝-L -σ b 3 y-b 2 z 2 -b 1 z 3
wherein b is i > 0, i=1, 2,3, for the system control coefficients such thatWherein P is z Is positive matrix, ++>
And 4, determining the value range of the parameter by utilizing the Lyapunov stability theory, well estimating the unknown state of the system, and globally stabilizing the closed-loop system, wherein the method comprises the following steps of:
firstly, selecting a Lyapunov function, then selecting an adjustable constant gain M on the basis of a dynamic compensator and an adaptive output feedback controller designed in the step 2 and the step 3 by utilizing the Lyapunov stability theory to ensure that the system is stable, namely, for the designed Lyapunov function V, the Earthway differential thereof meets the following conditionsWherein c 1 And c 2 Is a positive constant.
And 4.1, constructing a Lyapunov-Crohn's functional, determining the value range of the adjustable high gain M through the Lyapunov stability theory, and performing self-adaptive output feedback control on the single-link mechanical arm, wherein the method comprises the following steps of:
step 4.1.1, selecting Lyapunov-Crohn's Functions:
in the middle of
Solving the Itenus differential for the functional to obtain:
from the young's inequality:
according to hypothesis 3, there are:
step 4.1.2, selectThe inequality mentioned above is brought to +.>Can be obtained by:
wherein γ=1- (3+2 ρ) 1 )b 3 ‖P z ‖，m 4 ＝4‖P z ‖ 2 +4‖P z ‖ 2 ‖B 1 ‖ 2 +2‖P z ‖‖B 1 ‖，m 6 ＝m 4 +4‖P‖ 2 ‖B 1 ‖ 2 ，/>
And 4.1.3, controlling the self-adaptive output feedback of the single-link mechanical arm by using the determined parameter M.
The almost everywhere stability of the system was demonstrated using the anti-evidence method. Analysis may result in the system states e and z being almost everywhere bounded if the dynamic gain L is almost everywhere bounded, so it is only necessary to prove that L is almost everywhere bounded to derive that the system is almost everywhere globally bounded. Let lim be the rule of countercheck t→+∞ sup II L (t), e (t), z (t) II = + infinity. From this assumption, it can be derived when t < t f Any variable is almost everywhere bounded, so it is easy to find a finite time t 1 And a finite positive constant a such thatWhere inf represents the infinit. From the derivative of L, L being a non-decreasing function of initial value 1, there must be a finite time t 2 So that
At this time, the liquid crystal display device,for->At [ t ] 2 Infinity), is integrated on
Thus, L is almost everywhere globally bounded, and immediately the available system states e and z are almost everywhere bounded, so the closed loop system is almost everywhere globally bounded.
The effectiveness of the design method in the invention is verified by a single-link mechanical arm model.
Assume that the values of the parameters are d=1, n=6, b=1, m=0.5, h=0.5, k m =1, further convertible into:
dx 1 ＝x 2 dt
dx 2 ＝x 3 dt-(0.5x 2 +3sin(x 1 ))dt+0.5x 1 (t-d(t))dω
dx 3 ＝udt-(x 2 +x 3 )dt
wherein the nonlinear function phi 1 ＝0，φ 2 ＝-(0.5x 2 +3sin(x 1 ))，/>φ 3 ＝-(x 2 +x 3 )，/>Select θ 1 ＝3，θ 2 =1 to satisfy hypothesis 1.
The dynamic compensator is designed as
The coordinates are converted into:
the control law is as follows:
u＝L 3+σ M 3 v＝-L 2+σ M 3 b 3 y-L 3+σ M 3 b 2 z 2 -L 3+σ M 3 b 1 z 3
the relevant parameters are selected as follows: a, a 1 ＝1.5，a 2 ＝2，a 3 ＝1，σ＝0.4，M＝2.32，b 1 ＝0.5，b 2 ＝3.1，b 3 The system output y (t) is shown in fig. 4, the adaptive dynamic adjustable parameter L of the system is shown in fig. 5, the time curve of the system (t) is shown in fig. 5, and the error e of the system compensator 1 (t)，e 2 (t)，e 3 (t) time-dependent curve as shown in FIG. 6, system state x 1 (t)，x 2 (t)，x 3 The time-dependent curve of (t) is shown in fig. 7, and the time-dependent curve of the system control input u (t) is shown in fig. 8, from which it can be seen that the system can remain stable and the observer can well estimate the unknown state of the system under the action of the controller designed in this embodiment.
Claims (2)
1. The self-adaptive output feedback control method for the single-link mechanical arm is characterized by comprising the following steps of:
step 1, modeling a single-link mechanical arm to obtain an uncertain nonlinear system model with random disturbance influence, wherein uncertain factors comprise an unknown growth rate, an unknown control direction, an unknown output and an unknown time lag;
step 2, aiming at the system state which cannot be measured, designing a dynamic compensator to estimate the system state;
step 3, designing a self-adaptive output feedback controller based on the estimated state and system output;
step 4, determining the range of an adjustable constant gain M introduced in coordinate transformation by utilizing the Lyapunov stability theory, and controlling the single-link mechanical arm;
the system model established in the step 1 is specifically as follows:
step 1.1, the system dynamics of a single-link mechanical arm are as follows:
wherein q is,Respectively representing the position, the speed and the acceleration of the connecting rod; τ r Torque generated for the electrical system; u is a control input for representing an electromechanical torque; />Random disturbance ω with time-varying delay d (t) and torque defined in the system; d is mechanical inertia; b is the viscous friction coefficient of the joint; n is a positive constant related to load mass and gravity coefficient; m is armature inductance; h is the armature resistance; k (K) m Is the back electromotive force coefficient;
order theModeling a single-link mechanical arm as:
further convert into:
dx 1 ＝x 2 dt
y＝h(x 1 )
the above system is represented as a special class of systems, namely random nonlinear time-lapse systems with unknown output functions:
y＝h(x 1 )
wherein x is 1 ,x 2 And x 3 In order to be in the state of the system,u and y are the input and output of the system, respectively; unknown function phi i (. Cndot.) and->A continuous nonlinear function with an initial value of 0; h (·) is the output function of the system; d (t) is a time lag link of the system; omega is probability space +.>Standard wiener process;
step 1.2, for the system model established in step 1.1, assume the following:
assuming 1, the presence of unknown positive constants d and τ causes the time-varying time-lag d (t) to satisfy 0<d(t)<d and
suppose 2, for an unknown nonlinear function in the system i (. Cndot.) A. Cndot. CPresence of unknown positive constant θ 1 And theta 2 So that the following formula holds:
in the formula, |x| | represents the norm;
assuming that 3, the continuous output function h(s) satisfies h (0) =0,and there is a constant ρ 1 Not less than 0 and ρ 2 More than or equal to 0 so that |h(s) |is less than or equal to ρ 1 s+ρ 2 ；
The dynamic compensator in the step 2 is specifically designed as follows:
step 2.1, designing a dynamic compensator based on the system model established in the step 1.1, wherein the dynamic compensator comprises the following steps:
in the method, in the process of the invention,for compensator state, L is dynamic high gain and L (0) =1, parameter a i >0, i=1, 2,3, there is l>0 is A T P+PA is less than or equal to- (l+2) I, wherein P is positive definite matrix, and ∈A>
2.2, carrying out the following coordinate transformation on the system model established in the step 1.1:
in e i I=1, 2,3, representing an estimation error; z i I=1, 2,3, representing a state after the system coordinate transformation; v represents input after system coordinate transformation, M is more than or equal to 1, adjustable normal gain is achieved, and sigma is more than or equal to 0 and is any constant;
the system transforms into the following form:
wherein the method comprises the steps of
e＝[e 1 ,e 2 ,e 3 ] T ,z＝[z 1 ,z 2 ,z 3 ] T ,B 1 ＝[a 1 ,a 2 ,a 3 ] T ,B 2 ＝[0,0,1] T ,B 3 ＝[1,0,0] T ,
The design adaptive output feedback controller described in step 3 is in the form of:
v＝-L -σ b 3 y-b 2 z 2 -b 1 z 3
wherein b is i >0, i=1, 2,3, for the system control coefficients such thatWherein P is z Is a positive definite matrix of the matrix and the matrix,
the step 4 specifically comprises the following steps:
and 4.1, constructing a Lyapunov-Crohn's functional, determining the value range of the adjustable high gain M through the Lyapunov stability theory, and performing self-adaptive output feedback control on the single-link mechanical arm, wherein the method comprises the following steps of:
step 4.1.1, selecting Lyapunov-Crohn's Functions:
in the middle of
Solving the functional functions to obtain the differential of the Italis by solving the functional functions:
from the young's inequality:
according to hypothesis 3, there are:
step 4.1.2, selectThe inequality mentioned above is brought to +.>Is obtained by:
wherein γ=1- (3+2 ρ) 1 )b 3 ||P z ||，m 4 ＝4||P z || 2 +4||P z || 2 ||B 1 || 2 +2||P z ||||B 1 ||，m 6 ＝m 4 +4||P|| 2 ||B 1 || 2 ，/>
Step 4.1.3, controlling the self-adaptive output feedback of the single-link mechanical arm by utilizing the determined parameter M;
the system almost everywhere stability is proved by an anti-evidence method, and if the dynamic gain L is almost everywhere bounded, the system states e and z are almost everywhere bounded, so that the system is almost everywhere global bounded only by proving the almost everywhere bounded of the L; let lim be the rule of countercheck t→+∞ sup L (t), e (t), z (t) | the process is carried out in a manner of = +++, from this assumption, we find when t<t f Any variable is almost everywhere bounded, and therefore, a finite time t is found 1 And a finite positive constant a such thatWherein inf represents the infinit; from the derivative of L, L being a non-decreasing function of initial value 1, there must be a finite time t 2 So that
At this time, the liquid crystal display device,for->At [ t ] 2 , + -infinity) on integral to obtain
2. The adaptive output feedback control method for a single link mechanical arm according to claim 1, wherein the selecting method of the adjustable constant gain M in step 4 is as follows:
firstly, selecting a Lyapunov function, then selecting an adjustable constant gain M on the basis of a dynamic compensator and an adaptive output feedback controller designed in the step 2 and the step 3 by utilizing the Lyapunov stability theory to ensure that the system is stable, namely, for the designed Lyapunov function V, the Earthway differential thereof meets the following conditionsWherein c 1 And c 2 Is a positive constant.
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| CN113433825A (en) * | 2021-06-22 | 2021-09-24 | 广州大学 | Self-adaptive fault-tolerant control method and system of single-link mechanical arm and storage medium |
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