CN117215240A - Pre-allocation time preset performance control method of nonlinear electromechanical servo system - Google Patents
Pre-allocation time preset performance control method of nonlinear electromechanical servo system Download PDFInfo
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Abstract
The invention discloses a pre-allocation time preset performance control method of a nonlinear electromechanical servo system, which comprises the following steps: acquiring a state equation of a nonlinear system with state constraint of an electromechanical servo system; determining a control target according to the state equation and the state constraint condition; designing error variables according to a state equation, performing recursive design based on a back-step control method, and designing virtual control variables and actual control variables according to a control target; and constructing a preset performance barrier Lyapunov function V, substituting the designed virtual control variable and the actual control input, and verifying whether the virtual control variable and the actual control input meet the pre-allocation time stability criterion. The control scheme is proved to be capable of achieving the stability of the closed-loop system in the pre-allocation time and tracking the given track signal on the premise of meeting the preset performance index. Compared with the prior art, the method and the device can ensure that the control track of the whole closed-loop system is globally converged in the pre-designated time according to the parameter condition of the electromechanical system model, and meet the pre-set performance index.
Description
Technical Field
The invention relates to the field of pre-allocation preset performance time control, in particular to a pre-allocation time preset performance control method of a nonlinear electromechanical servo system.
Background
With the advent of the 4.0 era of industry, servo systems are increasingly used in manufacturing activities in various industries to realize high-quality and high-precision product manufacturing, and the shift from traditional fields of spinning, packaging, printing and the like to emerging fields of electronic equipment manufacturing, industrial robots and the like is realized. However, the existing motor is difficult to simultaneously meet the requirements of flexibility, quick response, impact resistance and the like, the breakthrough of motor technology is a key link for realizing the commercialized application of the robot in the future, and huge technology iteration space exists.
Due to the continuous improvement of the control requirements of factors such as friction, external interference, quick response and the like existing in the electromechanical servo system, the motor parameters, the movement speed and the tracking error of the electromechanical servo system are required to be limited in order to ensure the stability, the robustness and the control precision of the control system. Therefore, it is of great practical importance to study the constraint control of the servo system. Secondly, in the modeling process of the traditional nonlinear electromechanical servo system, the convergence time of the system is influenced by initial conditions and design parameters, so that the random setting of the convergence time of the electromechanical servo system is difficult to realize. How to solve the pre-allocation time control method of the nonlinear electromechanical servo system with predefined performance indexes and state constraints becomes a technical problem to be solved.
Disclosure of Invention
The invention aims to: in order to overcome the defects in the prior art, the invention provides a pre-allocation time preset performance control method of a nonlinear electromechanical servo system, which ensures that the tracking control track of the nonlinear electromechanical servo system meets preset performance indexes in the pre-allocation time according to preset performance control, pre-allocation time control and a back-step recursion design, thereby enhancing the control precision and the rapid stability of the nonlinear electromechanical servo system.
The technical scheme is as follows: the invention discloses a pre-allocation time preset performance control method of a nonlinear electromechanical servo system, which is based on nonlinear system global preset time control with state constraint and a preset performance control strategy, and comprises the following steps:
step 1: describing a dynamics equation of a nonlinear electromechanical servo system by using an Euler-Lagrange dynamics model, and obtaining a state equation of the nonlinear system with state constraint according to the dynamics equation;
step 2: according to the state equation and the state constraint condition, determining that the control target of the designed control scheme is: the control track of the whole mechanical arm closed-loop system is guaranteed to be globally converged in a preset time and meets a preset performance index;
step 3: adopting a method of combining preset time control and preset performance control of a nonlinear system, and constructing a new preset performance BLF by combining a Barrier Lyapunov Function (BLF) method to perform control design; designing error variables according to a state equation of the nonlinear system, performing recursive design based on a backstepping control method, and designing virtual control variables and actual control variables according to a control target; the error variable, the virtual control variable and the actual control input to be designed according to the control target are specifically as follows:
1) The design error variables are:
e 1 ＝x 1 -y r ,
e i ＝x i -α i-1 ,
wherein e 1 ,e i Is the error variable alpha i-1 Is a virtual controller, y r Is the desired trajectory and satisfiesC d,k > 0 is a constant, k=0, 1, …, n;
2) The design virtual control variables and the actual control inputs are:
wherein->
Wherein->
Wherein->
Wherein->
Wherein, the function h (t): R→ {0,1} is defined as h (t):to simplify the notation, we will omit the parameters of the function, using h to refer to h (t); alpha 1 ,α 2 ,α i U is the input of the first virtual control, the second virtual control, the ith virtual control and the actual control respectively; f (f) 1 ,f 2 ,f i ,f n Are all known smooth functions; />As a time-varying function, t p For pre-allocation time, +.>Is constant and satisfies->The time-varying gain is +.>ρ i (t) is a predefined performance function, +.>The first derivatives, beta, of the squares of the sum of squares of the predefined performance functions, respectively i ,D i All the values are more than 0, tau is a positive integer and 2τ is more than or equal to n;
step 4: constructing a proper predetermined performance obstacle Lyapunov function V, deriving the function V, substituting the designed virtual control variable and actual control input, and verifying whether the designed virtual control variable and actual control input meet the pre-allocation time stability criterionWherein->As a time-varying function; if the virtual control variable and the actual control input are satisfied, the designed control strategy can realize the global pre-allocation time stabilization of the system, and if the virtual control variable and the actual control input are not satisfied, the virtual control variable and the actual control input are required to be adjusted, and the virtual control variable and the actual control input are carried into the calculation again until an inequality is established;
step 5: according to Lyapunov stability theory, stability analysis is carried out on the global pre-allocation time convergence control scheme designed in the steps 1 to 4, and the method is proved to be capable of guaranteeing that all variables in a closed loop system are bounded in pre-allocation time, and meanwhile tracking errors can meet preset performance indexes in preset time.
Further, in the step 1, the state equation of the nonlinear system with state constraint is expressed as:
the nonlinear electromechanical servo system dynamics model is as follows:
wherein,j represents rotor inertia, < >>Respectively the angular position, the angular speed and the angular acceleration of the motor, M and M 0 Link mass and load mass, L 0 ,R 0 Respectively representing the length of the connecting rod and the radius of the load, wherein I, L and R are respectively armature current, armature inductance and armature resistance, G and B 0 ,K τ ,K B Respectively represent a gravity coefficient, a viscous friction coefficient, a conversion coefficient and a back electromotive force coefficient, V 0 Representing an input voltage;
defining state variablesThe state equation of the nonlinear system with state constraint is obtained according to the dynamics model equation (1):
wherein,representing the electromechanical system model error.
Further, in the step 2, the control target is:
target 1: all signals in the closed loop system are globally pre-assigned time stable, wherein the time t is pre-determined p Arbitrarily selecting according to the requirements of a designer;
target 2: the output signal y (t) is at a pre-specified time t p Inner implementation desired trajectory y r Pre-specified tracking of (t) independent of any system parameters or initial conditions;
target 3: the desired predetermined performance index is met while ensuring that all state variables do not violate constraints.
Further, in the step 4, the method specifically includes the following steps:
(41) The predefined performance functions constructed are:
wherein ρ is r,0 ,ρ r,∞ And l r Is a positive parameter with an initial value of ρ r (0)＝ρ r,0 +ρ r,∞ The method comprises the steps of carrying out a first treatment on the surface of the r is a positive constant of 1 to n;
(42) Combining the barrier Liapunov function and the predefined performance function, the following Lyapunov function V is designed as follows:
wherein the tracking error e i All error variables, including (t), are limited to a region bounded by a predefined performance function, i.e. |e i (t)|＜ρ i (t)；
(43) Based on the state equation and e of the nonlinear system with state constraint 1 ＝x 1 -y r Calculating an error variable e 1 Is obtained by:
the function h (t) is designed as R→ {0,1 }:
design of the 1 st virtual control variable alpha 1 The method comprises the following steps:
wherein,is the normal number to be designed;
the Lyapunov function was designed as follows:
when t is more than or equal to 0 and less than t p In this case, V is calculated 1 As regards the derivative of time, it is possible to obtain:
due toThus, formula (5) is rewritable:
wherein the auxiliary variable
(44) Based on the state equation and e of the nonlinear system with state constraint 2 ＝x 2 -α 1 Calculating an error variable e 2 Is obtained by:
design of the 2 nd virtual control variable alpha 2 The method comprises the following steps:
wherein,is the normal number to be designed;
from equation (3), we can derive the time derivative of the first virtual control variableExpressed as:
the second positive Lyapunov function was designed as follows:
when t is more than or equal to 0 and less than t p At the time of V 2 Conduct derivation taking into accountAnd bringing formulae (7) - (9) into availability:
wherein the auxiliary variable
(45) For V i Taking the derivative with respect to time t, calculated by the equation above,expressed as:
wherein the auxiliary variable
(46) For V n Taking the derivative with respect to time t, calculated by the equation above,expressed as:
wherein the auxiliary variable
Further, the analyzing and verifying process in the step 5 includes:
the Lyapunov function of the whole closed loop system is:
according to the definition and nature of the log function, there are:
the derivative of V with respect to time can be obtained
When t is more than or equal to 0 and less than t p The computational reduction (16) can be obtained:
from equation (17), it can be seen that e is the Lyapunov function V (t) →0 selected when t→infinity i (t)→0；
From the formulas (14) and (17), it is possible to obtain:
due to e 1 (t) is continuous, and when t.fwdarw.infinity, we obtainBy continuously performing the lobida algorithm on the alpha, alpha can be known 1 Is C n-1 Class functions; similarly, a is derived i Is C n-i Class functions.
Further, all signals e of the closed loop system 1 ,...,e n ,α 1 ,…α n U are all globally pre-allocation time bounded, combined with a designed predefined performance function ρ r (t) analysis, system output y (t) and reference trajectory y r Tracking error e between (t) 1 Converging to a value of + - ρ within a prescribed time period 1 Pre-assigning function domains for upper and lower bounds; taking into account the global convergence of the closed loop system and the nature of the pre-allocation performance function, the inequality |e is derived i (t)|＜ρ i (t) is true, the reference trajectory satisfies|y r |≤C d,0 Let |x 1 |＝|e 1 +y r |≤|e 1 |+|y r |≤ρ 1,0 +ρ 1,∞ +C d,0 ＝C x1 Thereby ensuring state constraintWill not be violated; similarly, get e i ,α i-1 Is bounded and x i The state constraint is not violated, where i=2, …, n.
The beneficial effects are that: the invention provides a pre-allocation time preset performance control strategy for a nonlinear system, which combines a predefined performance function and a log-type Barrier Liapunov Function (BLF), and designs a preset performance BLF for restraining a state error, so that the tracking error is converged into a pre-allocation function domain within a specified time, and the system is ensured to meet a state constraint condition. The proposed strategy allows the convergence time to be arbitrarily set to accommodate system operating requirements, independent of initial conditions and design parameters. Furthermore, the proposed control scheme not only relaxes the requirements of the state scaling transformation, but also allows closed system operation over any pre-allocation time interval.
Drawings
FIG. 1 is a schematic diagram of an electromechanical system according to an embodiment of the present invention;
FIG. 2 is a flowchart of a pre-allocation time control method according to an embodiment of the present invention;
FIG. 3 shows the output y and the expected output y of the present invention r ConstraintIs a curve of (2);
FIG. 4 is a state variable x of the present invention 2 And its constraintIs a curve of (2);
FIG. 5 shows the tracking error e of the present invention 1 Constraint ρ 1 Is a curve of (2);
FIG. 6 shows the tracking error e of the present invention 2 Constraint ρ 2 Is a curve of (2);
fig. 7 is a graph of the control input u of the present invention.
Detailed Description
The technical scheme of the invention is described in detail below.
The invention discloses a pre-allocation time preset performance control method of a nonlinear electromechanical servo system, which takes the electromechanical servo system as an example in the embodiment to carry out control strategy design, wherein a dynamics model of the nonlinear electromechanical servo system is as follows:
wherein,j represents rotor inertia, < >>Respectively the angular position, the angular speed and the angular acceleration of the motor, M and M 0 Link mass and load mass, L 0 ,R 0 Respectively representing the length of the connecting rod and the radius of the load, wherein I, L and R are respectively armature current, armature inductance and armature resistance, G and B 0 ,K τ ,K B Respectively represent a gravity coefficient, a viscous friction coefficient, a conversion coefficient and a back electromotive force coefficient, V 0 Representing the input voltage.
Defining state variablesThe conversion of the kinetic equation (1) can be obtained:
wherein,and the model error of the electromechanical system is represented, y is the control output of the system, and the angular position, the angular speed and the angular acceleration of the motor can only operate in a certain range because the motor is limited by the physical characteristics of the motor. Thus, assume that the state variables of the system are constrained to an open set +.>In (1)/(2)>I=1, 2,3 for normal number. The system for the invention is a more generalized nonlinear system, so that models meeting the requirements of the nonlinear system with state constraint can use the pre-allocation time preset performance control strategy provided by the invention, and therefore, the strategy is also applicable to actual models such as nonlinear electromechanical servo systems with wider required state parameters according to the actual production requirements.
Based on a more general nonlinear system as a research object, a control method of an electromechanical servo system is designed, and a state equation of the nonlinear system with state constraint comprises:
wherein the system state is expressed asAnd->u epsilon R and y epsilon R are respectively the control input and the control output of the system; />Is a known smooth continuous function; all state variables are limited to an open set +.>In (I)>Is a known positive number, and the number of the positive numbers is, i=1, 2,..n.
Obviously, the dynamics model equation (2) of the nonlinear electromechanical servo system is a special case of the state equation (3) when i=2, and all the features of the state equation (3) are satisfied. Therefore, the pre-allocation time scheduled performance control strategy of the nonlinear system with state constraints proposed by the present invention can be used to deal with the control problem of electromechanical servo systems.
The pre-allocation time preset performance control of the nonlinear electromechanical servo system with state constraint is realized by adopting the control method, and the specific steps are as follows:
(1) Constructing a design error variable according to a state equation of the nonlinear system, wherein the design error variable is as follows:
wherein e 1 ,e i Is the error variable alpha i-1 Is a virtual controller, y r Is the desired trajectory and satisfiesC d,k > 0 is a constant, k=0, 1, …, n.
(2) The recursive design is carried out based on a backstepping control method, virtual control variables and actual control inputs are designed according to a control target, and the method comprises the following steps:
wherein, the function h (t): R→ {0,1} is defined as h (t):to simplify the notation, we will omit the parameters of the function, using h to refer to h (t); alpha 1 ,α 2 ,α i U is the firstA virtual control, a second virtual control, an ith virtual control, and an actual control input; f (f) 1 ,f 2 ,f i ,f n Are all known smooth functions; />As a time-varying function, t p For pre-allocation time, +.>Is constant and satisfies->Time-varying gain->ρ i (t) is a predefined performance function, +.>The first derivatives, beta, of the squares of the sum of squares of the predefined performance functions, respectively i ,D i And (3) the design parameters are more than 0, tau is a positive integer, and 2tau is more than or equal to n.
(3) Constructing a proper predetermined performance obstacle Lyapunov function V, deriving the function V, substituting the designed virtual control variable and actual control input, and verifying whether the designed virtual control variable and actual control input meet the pre-allocation time stability criterionWherein the method comprises the steps ofAs a time-varying function; if the virtual control variable and the actual control input are satisfied, the designed control strategy can realize the global preallocation time stabilization of the system, and if the virtual control variable and the actual control input are not satisfied, the virtual control variable and the actual control input are required to be adjusted, and the virtual control variable and the actual control input are carried into the calculation again until the inequality is satisfied.
(41) The predefined performance functions constructed are:
wherein ρ is r,0 ,ρ r,∞ And l r Is a positive parameter with an initial value of ρ r (0)＝ρ r,0 +ρ r,∞ R is a positive constant of 1 to n.
(42) Combining the barrier Liapunov function and the predefined performance function, the following Lyapunov function V is designed as follows:
wherein the tracking error e i All error variables, including (t), are limited to a region bounded by a predefined performance function, i.e. |e i (t)|＜ρ i (t)。
The main control objectives of the present invention are:
target 1: all signals in the closed loop system are globally pre-assigned time stable, wherein the time t is pre-determined p May be arbitrarily selected according to the requirements of the designer.
Target 2: the output signal y (t) is at a pre-specified time t p Inner implementation desired trajectory y r Pre-specified tracking of (t) which is not affected by any system parameters or initial conditions.
Target 3: the desired predetermined performance index may be met while ensuring that all state variables do not violate constraints.
Before introducing the main design process, the assumption should be explained first
Specific steps of a nonlinear system pre-allocation time preset performance control scheme with state constraint are given below, and the design steps are divided into n steps.
Step 1: based on the state equation and e of the nonlinear system with state constraint 1 ＝x 1 -y r Calculating an error variable e 1 Is obtained by:
the function h (t) is designed as R→ {0,1 }:
design of the 1 st virtual control variable alpha 1 The method comprises the following steps:
wherein,is the normal number to be designed;
the Lyapunov function was designed as follows:
when t is more than or equal to 0 and less than t p In this case, V is calculated 1 As regards the derivative of time, it is possible to obtain:
due toThus, formula (12) can be rewritten as:
wherein the auxiliary variable
Step 2: according to non-linear systems with state constraintsEquation of state and e 2 ＝x 2 -α 1 Calculating an error variable e 2 Is obtained by:
design of the 2 nd virtual control variable alpha 2 The method comprises the following steps:
wherein,is the normal number to be designed;
from equation (10), we can derive the time derivative of the first virtual control variableExpressed as:
the 2 nd positive definite Lyapunov function was designed as follows:
when t is more than or equal to 0 and less than t p At the time of V 2 Conduct derivation taking into accountAnd bringing formulae (14), (15) into availability: />
Wherein the auxiliary variable
Step i (i is more than or equal to 2 and less than or equal to n-1): according to the state equations (3) and e i ＝x i -α i-1 Calculating an error variable e i Is obtained by:
design of the ith virtual control variable alpha i The method comprises the following steps:
wherein,is the normal number to be designed.
The i-th positive definite Lyapunov function is designed as follows:
when t is more than or equal to 0 and less than t p At the time, calculate V i The derivative with respect to time, and substituting (19), (20), can be obtained:
due toThus, formula (22) may be rewritten as: />
Wherein the auxiliary variable
And (n) step: according to the state equations (3) and e n ＝x n -α n-1 Calculating an error variable e n Is obtained by:
the actual controller u is designed as follows:
wherein,is the normal number to be designed.
The n-th positive definite Lyapunov function was designed as follows:
when t is more than or equal to 0 and less than t p At the time, calculate V n The derivative with respect to time, and substituting (24), (25), can be obtained:
due toThus, formula (27) may be rewritten as:
wherein the auxiliary variable
(4) According to Lyapunov stability theory, stability analysis is carried out on a designed global pre-allocation time convergence control scheme, and the control method provided by the invention is proved to be capable of ensuring that all variables in a closed loop system are bounded in pre-allocation time, and meanwhile, tracking errors can meet preset performance indexes in preset time.
Considering a nonlinear system (3) with state constraints, if a virtual control (10), (15), (20), an actual controller (25) is chosen, in combination with the barrier lispro function, a predefined performance function, the following characteristics can be guaranteed: tracking error e 1 (t) at a pre-specified time t p The internal global convergence is zero, i.eSimultaneously meeting predefined performance indexes; virtual control variable alpha i Is C n-i The class function, i=1, 2, n; all signals of the closed loop system are bounded during the pre-allocation time and when t.gtoreq.t p Time e i 0.ident; all state variables of the system do not violate the state constraint, i.e. guarantee +.>
And (3) proving:
considering the Lyapunov function of a closed loop system as
According to the definition and nature of the log function, there are:
the derivative of V with respect to time can be obtained
When t is more than or equal to 0 and less than t p The computational reduction (31) can be obtained:
from equation (32), it can be seen that e is the Lyapunov function V (t) →0 selected when t→infinity i (t)→0。
From the formulas (29) and (32), it is possible to obtain:
due to e 1 (t) is continuous, and when t.fwdarw.infinity, we obtainBy continuously performing the lobida algorithm on the alpha, alpha can be known 1 Is C n-1 Class functions; similarly, a is derived i Is C n-i Class functions.
From all the above-mentioned derivation steps, all the signals e of the closed loop system are known 1 ,...,e n ,α 1 ,…α n U is globally pre-assigned time bounded when t is greater than or equal to t p In the time-course of which the first and second contact surfaces,due to V n Is continuous and non-positive, can be inferred that V n =0, thus e i 0.ident; predefined performance function ρ of a joint design r (t) analysis, system output y (t) and reference trajectory y r Tracking error e between (t) 1 Will converge to + - ρ within a prescribed time period 1 The function domains are pre-allocated for the upper and lower bounds. Taking into account the global convergence of the closed loop system and the nature of the pre-allocation performance function, the inequality |e is derived i (t)|＜ρ i (t) is true, the reference trajectory satisfies |y r |≤C d,0 Let |x 1 |＝|e 1 +y r |≤|e 1 |+|y r |≤ρ 1,0 +ρ 1,∞ +C d,0 ＝C x1 Thereby ensuring state constraint |x 1 |≤C x1 Is not violated. Similarly, get e i ,α i-1 Is bounded and x i The state constraint is not violated, where i=2, …, n.
Through the analysis, the control method provided by the invention can realize that the tracking control of the electromechanical servo system meets the preset performance indexes such as tracking error and maximum overshoot within the preset time, and does not violate the state constraint condition of the system. The pre-allocation time preset performance control strategy of the nonlinear electromechanical servo system with state constraint realizes all control targets.
Aiming at the control problem of a nonlinear system with state constraint, the invention provides a pre-allocation time preset performance control method. By controlling the design process, it can be found that in order to increase the negative certainty of the stable performance function, a time-varying gain term and an additional uncertainty term are introduced into the design, so that the setting time can be flexibly selected according to the pre-assigned index and tracked and controlled. The convergence time of the system is not only independent of design parameters, but also independent of initial conditions, and can be set according to our wish.
The proving process is to prove the effectiveness of the pre-allocation time preset performance control method provided by the invention in theory. The method provided by the invention is easier to apply in practical application. The main expression is as follows:
(1) From the characteristics of the predefined performance obstacle lyapunov function, the tracking error e i All error variables, including (t), are limited to an area bounded by a predefined performance function, i.e. |e i (t)|＜ρ i (t)。
(2) By setting ρ r,0 ,ρ r,∞ The overshoot and tracking error are preset, and the time constant t is adjusted p The convergence time of the system can be adjusted.
(3) The Matlab tool is used for applying the control strategy to the electromechanical system model, and simulation results show that the method designed by the invention can achieve the required control target.
System parameters were set to j=1.625×10 -3 kg·m 2 ,m＝0.506kg,M 0 ＝0.434kg,L 0 ＝0.305m,R 0 ＝0.023m,L＝0.025H,R＝0.5Ω,G＝9.8N/kg,B 0 ＝0.01625kg·m 2 ,K τ ＝K B The initial value of the state variable is set to x =0.9n·m/a 1 (0)＝x 2 (0) =0.01, the state constraint is set toSetting a predefined performance function to ρ 1 (t)＝(0.87-0.02)e -2t +0.02,ρ 2 (t)＝(1.98-0.05)e -2t +0.05, desired output curve y r (t) =0.5 cost, external disturbance d (t) =0.1 cost 12t, set convergence time t p =2s, in addition to that, beta 1 ＝0.5,β 2 ＝0.51,D 1 ＝D 2 ＝0.1,/>The main simulation results obtained are shown in fig. 3 to 7.
The foregoing embodiments are merely illustrative of the technical concept and features of the present invention, and are intended to enable those skilled in the art to understand the present invention and to implement the same, not to limit the scope of the present invention. All equivalent changes or modifications made according to the spirit of the present invention should be included in the scope of the present invention.
Claims (6)
1. A method for pre-allocation time scheduled performance control of a nonlinear electro-mechanical servo system, characterized by global scheduled time control and scheduled performance control strategy based on a nonlinear system with state constraints, comprising the steps of:
step 1: describing a dynamics equation of a nonlinear electromechanical servo system by using an Euler-Lagrange dynamics model, and obtaining a state equation of the nonlinear system with state constraint according to the dynamics equation;
step 2: according to the state equation and the state constraint condition, determining that the control target of the designed control scheme is: the control track of the whole mechanical arm closed-loop system is guaranteed to be globally converged in a preset time and meets a preset performance index;
step 3: adopting a method of combining preset time control and preset performance control of a nonlinear system, and constructing a new preset performance BLF by combining a Barrier Lyapunov Function (BLF) method to perform control design; designing error variables according to a state equation of the nonlinear system, performing recursive design based on a backstepping control method, and designing virtual control variables and actual control variables according to a control target; the error variable, the virtual control variable and the actual control input to be designed according to the control target are specifically as follows:
1) The design error variables are:
e 1 ＝x 1 -y r ,
e i ＝x i -α i-1 ,
wherein e 1 ,e i Is the error variable alpha i-1 Is a virtual controller, y r Is the desired trajectory and satisfiesC d,k > 0 is a constant, k=0, 1, …, n;
2) The design virtual control variables and the actual control inputs are:
wherein->
Wherein->
Wherein->
Wherein->
Wherein, the function h (t): R→ {0,1} is defined as h (t):to simplify the notation, we will omit the parameters of the function, using h to refer to h (t); alpha 1 ,α 2 ,α i U is the input of the first virtual control, the second virtual control, the ith virtual control and the actual control respectively; f (f) 1 ,f 2 ,f i ,f n Are all known smooth functions; />As a time-varying function, t p For pre-allocation time, +.>Is constant and satisfies->The time-varying gain is +.>ρ i (t) is a predefined performance function, +.>The first derivatives, beta, of the squares of the sum of squares of the predefined performance functions, respectively i ,D i All the values are more than 0, tau is a positive integer and 2τ is more than or equal to n;
step 4: constructing a proper predetermined performance obstacle Lyapunov function V, deriving the function V, substituting the designed virtual control variable and actual control input, and verifying whether the designed virtual control variable and actual control input meet the pre-allocation time stability criterionWherein the method comprises the steps ofAs a time-varying function; if the virtual control variable and the actual control input are satisfied, the designed control strategy can realize the global pre-allocation time stabilization of the system, and if the virtual control variable and the actual control input are not satisfied, the virtual control variable and the actual control input are required to be adjusted, and the virtual control variable and the actual control input are carried into the calculation again until an inequality is established;
step 5: according to Lyapunov stability theory, stability analysis is carried out on the global pre-allocation time convergence control scheme designed in the steps 1 to 4, and the method is proved to be capable of guaranteeing that all variables in a closed loop system are bounded in pre-allocation time, and meanwhile tracking errors can meet preset performance indexes in preset time.
2. The method for controlling the pre-distribution time reservation performance of a nonlinear electro-mechanical servo system according to claim 1, wherein in the step 1, a state equation of the nonlinear system having a state constraint is expressed as:
the nonlinear electromechanical servo system dynamics model is as follows:
wherein,j represents the mass of the rotor, q,respectively the angular position, the angular speed and the angular acceleration of the motor, M and M 0 Link mass and load mass, L 0 ,R 0 Respectively representing the length of the connecting rod and the radius of the load, wherein I, L and R are respectively armature current, armature inductance and armature resistance, G and B 0 ,K τ ,K B Respectively represent a gravity coefficient, a viscous friction coefficient, a conversion coefficient and a back electromotive force coefficient, V 0 Representing an input voltage;
defining a state variable x 1 ＝q,x 3 ＝I,u＝V 0 The state equation of the nonlinear system with state constraint is obtained according to the dynamics model equation (1):
wherein,representing the electromechanical system model error.
3. The method for controlling the pre-distribution time preset performance of the nonlinear electromechanical servo system according to claim 2, wherein in the step 2, the control target is:
target 1: in a closed loop systemThe signals are globally preassigned time-stable, with a predetermined time t p Arbitrarily selecting according to the requirements of a designer;
target 2: the output signal y (t) is at a pre-specified time t p Inner implementation desired trajectory y r Pre-specified tracking of (t) independent of any system parameters or initial conditions;
target 3: the desired predetermined performance index is met while ensuring that all state variables do not violate constraints.
4. The method for controlling the pre-distribution time preset performance of the nonlinear electromechanical servo system according to claim 1, wherein in the step 4, the method specifically comprises the following steps:
(41) The predefined performance functions constructed are:
wherein ρ is r,0 ,ρ r,∞ And l r Is a positive parameter with an initial value of ρ r (0)＝ρ r,0 +ρ r,∞ The method comprises the steps of carrying out a first treatment on the surface of the r is a positive constant of 1 to n;
(42) Combining the barrier Liapunov function and the predefined performance function, the following Lyapunov function V is designed as follows:
wherein the tracking error e i All error variables, including (t), are limited to a region bounded by a predefined performance function, i.e. |e i (t)|＜ρ i (t)；
(43) Based on the state equation and e of the nonlinear system with state constraint 1 ＝x 1 -y r Calculating an error variable e 1 Is obtained by:
the function h (t) is designed as R→ {0,1 }:
design of the 1 st virtual control variable alpha 1 The method comprises the following steps:
wherein,β 1 ,D 1 > 0 is the normal number to be designed;
the Lyapunov function was designed as follows:
when t is more than or equal to 0 and less than t p In this case, V is calculated 1 As regards the derivative of time, it is possible to obtain:
due toThus, formula (5) is rewritable:
wherein the auxiliary variable
(44) Based on the state equation and e of the nonlinear system with state constraint 2 ＝x 2 -α 1 Calculating an error variable e 2 Is obtained by:
design of the 2 nd virtual control variable alpha 2 The method comprises the following steps:
wherein,β 2 ,D 2 > 0 is the normal number to be designed;
from equation (3), we can derive the time derivative of the first virtual control variableExpressed as:
the second positive Lyapunov function was designed as follows:
when t is more than or equal to 0 and less than t p At the time of V 2 Conduct derivation taking into accountAnd bringing formulae (7) - (9) into availability:
wherein the auxiliary variable
(45) For V i Taking the derivative with respect to time t, calculated by the equation above,expressed as:
wherein the auxiliary variable
(46) For V n Taking the derivative with respect to time t, calculated by the equation above,expressed as:
wherein the auxiliary variable
5. The method for controlling the pre-distribution time reservation performance of a nonlinear electro-mechanical servo system according to claim 4, wherein the analyzing and verifying process in the step 5 includes:
the Lyapunov function of the whole closed loop system is:
according to the definition and nature of the log function, there are:
the derivative of V with respect to time can be obtained
When t is more than or equal to 0 and less than t p The computational reduction (16) can be obtained:
from equation (17), it can be seen that e is the Lyapunov function V (t) →0 selected when t→infinity i (t)→0；
From the formulas (14) and (17), it is possible to obtain:
due to e 1 (t) is continuous, and when t.fwdarw.infinity, we obtainBy continuously performing the lobida algorithm on the alpha, alpha can be known 1 Is C n-1 Class functions; similarly, a is derived i Is C n-i Class functions.
6. The method for pre-distribution time scheduled performance control of a nonlinear electro-mechanical servo system in accordance with claim 5, wherein all signals e of the closed loop system 1 ,...,e n ,α 1 ,...α n U are all globally pre-allocation time bounded, combined with a designed predefined performance function ρ r (t) analysis, system output y (t) and reference trajectory y r Tracking error e between (t) 1 Converging to a value of + - ρ within a prescribed time period 1 Pre-assigning function domains for upper and lower bounds; taking into account the global convergence of the closed loop system and the nature of the pre-allocation performance function, the inequality |e is derived i (t)|＜ρ i (t) is true, the reference trajectory satisfies |y r |≤C d,0 Let |x 1 |＝|e 1 +y r |≤|e 1 |+|y r |≤ρ 1,0 +ρ 1,∞ +C d,0 ＝C x1 Thereby ensuring state constraintWill not be violated; similarly, get e i ,α i-1 Is bounded and x i The state constraint is not violated, where i=2, …, n.
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