CN116382154A - Event trigger-based specified time high-precision control method - Google Patents
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Abstract
The invention discloses a high-precision control method for designated time based on event triggering, which comprises the following steps: step 1: designing a specified time high-precision controller based on a parameter Lyapunov equation; step 2: designing an event trigger mechanism based on a parameter Lyapunov equation to determine the update time t of a controller i The method comprises the steps of carrying out a first treatment on the surface of the Step 3: and (3) calculating to obtain an expression of the minimum trigger time interval according to the event trigger mechanism designed in the step (2). The method not only can realize the high-precision control target of the system at any appointed time, but also reduces the update times of the system actuator, so that the loss of the system actuator is greatly reduced, and the method has important engineering application value.
Description
Technical Field
The invention belongs to the field of high-precision control, relates to an event trigger control method, and in particular relates to an event trigger-based high-precision control method for designated time.
Background
As a typical representative of a high-precision motion system, a photolithography motion stage system needs to satisfy the motion performance requirements of high acceleration, high speed and high positioning precision by taking into account high rigidity and lightweight mechanical design technology. Meanwhile, with development of computer network technology and cross penetration, single-machine operation systems are less and less, and multi-machine collaborative operation complex systems are replaced. The introduction of standard serial communication interfaces and special local area network interfaces greatly enhances the interconnection capability among multiple computers and greatly accelerates the communication rate. Networking features have become one of the fundamental characteristics of servo systems of modern high precision instruments. Therefore, the event-triggering-based specified time control method has the advantages of high precision, ultra-fast response, communication resource saving, strong anti-interference capability and the like, and is not compatible with the increasingly complex performance requirements of the photoetching machine moving table system, and the development of related algorithm research of the method inevitably has very important application value.
Disclosure of Invention
In order to further improve the servo performance of the system in a network environment, the invention provides a high-precision control method for the designated time based on event triggering, which not only can realize the high-precision control target of the system at any designated time, but also reduces the update times of the actuator of the system, so that the loss of the actuator of the system is greatly reduced, and the method has important engineering application value.
The invention aims at realizing the following technical scheme:
a high-precision control method for specified time based on event triggering comprises the following steps:
step 1: designing a specified time high-precision controller based on a parameter Lyapunov equation:
wherein ,
at time t for system state i The state of time, i belongs to N, N is an integer set,/->
Is the parameter LySolution P of apunov equation at time t i Value of time of day->
Step 2: the following event trigger mechanism based on the parameter Lyapunov equation is designed to determine the update time t of the controller i ：
t i+1 ＝inf{t＞t i :βγx T Px-f(t)≤0}；
wherein ,
e is the measurement error, and is the error in the measurement,
wherein the initial value of gamma is +.>
γ * Is at a time L sufficiently close to L * The value of the time gamma, L is any given time, L * < L, beta.E (0, 1) is a designed parameter,
α 2 ＝2dλ max (P n ) N represents the dimension of the system state,
E n ＝diag(n-1,n-2,…,1,0)，P n p (1) is the value of the solution P of the parametric Lyapunov equation when γ=1, λ max Max, diag represent the maximum eigenvalue, maximum and diagonal matrix, c, respectively kj ≥0，k＝1,2,…,n,j＝1,2…,k；
Step 3: according to the event triggering mechanism designed in the step 2, calculating to obtain the following expression of the minimum triggering time interval:
compared with the prior art, the invention has the following advantages:
1. the event trigger control method based on the parameter Lyapunov equation can eliminate the influence of the system matrix on the minimum trigger time interval, simultaneously establishes an expression of the minimum trigger time interval, has clear and simple relation with the design parameters, and facilitates the control performance adjustment of the controlled system.
2. According to the method provided by the invention, by designing the high-precision controller based on the appointed time triggered by the event, the high-precision control target of the system can be realized at any appointed time, and the update times of the system executor are reduced, so that the loss of the system executor is greatly reduced.
Drawings
Fig. 1 is a flow chart of the method for specifying time and high precision based on the event trigger mechanism.
FIG. 2 shows a state of the system, namely x, in an embodiment of the invention 1 。
Detailed Description
The following description of the present invention is provided with reference to the accompanying drawings, but is not limited to the following description, and any modifications or equivalent substitutions of the present invention should be included in the scope of the present invention without departing from the spirit and scope of the present invention.
The invention provides a high-precision control method for designated time based on event triggering, as shown in fig. 1, comprising the following steps:
step 1: the method for designing the specified time high-precision controller based on the parameter Lyapunov equation comprises the following specific steps:
step 11: the nonlinear system model considered can be expressed as:
where x is a state variable of the system,
for the derivative of the state variable, u is the control input to the system, and the state matrix for the matrix A and b systems is in the specific form as follows:
Step 12: the method comprises the steps of designing a specified time high-precision controller based on a parameter Lyapunov equation, wherein the specified time high-precision controller is specifically expressed as:
wherein T of the upper corner mark is the matrix transpose, e.g. b T Represents the transpose of b,
at time t for system state i The state of the moment i belongs to N, where N is an integer set, +.>
The solution P to the parametric Lyapunov equation is at time t as follows i Value of time of day
A T P+PA-Pbb T P＝-γP (3)；
wherein
Gamma is a time-varying parameter where L is any specified time and t e 0, L. Its initial value gamma 0 The method comprises the following steps:
wherein beta epsilon (0, 1) is a design parameter,
α 2 ＝2dλ max (P n ) Here, where
E n ＝diag(n-1,n-2,…,1,0)，P n The value P of the solution P of the parameter Lyapunov equation (2) when γ=1 is =p (1), λ max Max, diag represent the maximum eigenvalue, maximum value, and diagonal matrix of the matrix, respectively.
Step 2: designing an event trigger mechanism based on a parameter Lyapunov equation to determine the update time t of a controller i The method comprises the following specific steps:
step 21: before designing the trigger mechanism, the following measurement error e is first defined:
step 22: the following event trigger mechanism is designed:
t i+1 ＝inf{t＞t i :βγx T Px-f(t)≤0} (5)；
wherein 
Looking at the definition of γ (i.e., equation (4)), it is known that γ tends to be positive to infinity at time L. The P (gamma) approach to positive infinity; feedback controllers based on the event triggering mechanism of equations (2) and (5) are therefore physically unrealizable. In addition, gamma is in the time zone t epsilon L, + -infinity) is not defined in the text, causing the controller (2) to operate in the time zone t e L, + -infinity) also in are not defined. In order to design a physically realizable controller, a gamma design method is given:
wherein ,γ* Is a constant large enough to be a time L close enough to L * (L * < L) time γ. In addition, when t > L * When due to
Thus P e ＝0,f(t)＝x T P γ* BB T P γ* e and the event trigger mechanism becomes:
t i+1 ＝inf{t＞t i :βγ * x T P γ* x-2e T P γ* bb T P γ* e}；
here P γ* Is when γ=γ in equation (3) * The only positive solution at that time.
Step 3: according to the event trigger mechanism designed in the step 2, calculating to obtain the minimum trigger time interval tau e Is an expression of (2).
Step 31: obtaining an event trigger time interval function according to the event trigger mechanism designed in the step 2
The minimum triggering time interval is the value of the event triggering time interval function S from 0 to +.>
The time required.
Step 32: deriving the event trigger time interval function S, and obtaining:
step 33: the minimum trigger time interval of the designed event trigger mechanism is obtained according to the comparison quotation:
examples:
in the embodiment, the control of the single-link manipulator system is taken as a specific implementation mode, and the control gain is designed by adopting the method of the invention. The method specifically comprises the following steps:
step 1: the definition conversion is utilized to convert the single-link manipulator system into the nonlinear system model in the general form, and the specific process is as follows:
consider the relative dynamics model of a single link manipulator system as follows:
wherein ,q1 ,q 2 Is the angular position, I and J are moment of inertia, k is the spring rate, M is the total mass, L is the distance, and u is the moment of rotation input, i.e., the control input.
Definition of the definition
And MgL =i=j=10 and k=1, the single link manipulator system can be written as follows:
after which x= [ x ] is defined 1 ,x 2 ,x 3 ,x 4 ]This system can be written in the form of a system (1), i.e
Wherein x is a state variable of the system, u is a control input of the system, and the specific forms of the state matrixes of the matrix A and the matrix b are as follows:
wherein 
Represents a nonlinear term and satisfies
Here c kj 0, where k=1, 2, …, n, j=1, 2 …, k, n=4 represents the dimension of the system state.
Then, a high-precision controller based on an event trigger mechanism is designed, and the high-precision controller is specifically expressed as follows:
wherein T of the upper corner mark is a matrix transpose such as b T Represents the transpose of b,
at time t for system state i The state of the moment i belongs to N, where N is an integer set, +.>
The solution P to the parametric Lyapunov equation is at time t as follows i Value of time of day
A T P+PA-Pbb T P＝-γP (3)；
wherein
Gamma is a time-varying parameter where L is any specified time and t e 0, L. Its initial value gamma 0 The method comprises the following steps:
wherein beta epsilon (0, 1) is a design parameter,
α 2 ＝2dλ max (P n ) Here, where
E n ＝diag(n-1,n-2,…,1,0)，P n The value P of the solution P of the parameter Lyapunov equation (2) when γ=1 is =p (1), λ max Max, diag represent the maximum eigenvalue, maximum and diagonal matrix of the matrix, respectively.
Step 2: designing an event trigger mechanism based on a parameter Lyapunov equation to determine the update time t of a controller i 。
Before designing the trigger mechanism, the measurement error e is first defined as follows:
the event trigger mechanism is then designed as follows:
t i+1 ＝inf{t＞t i :βγx T Px-f(t)≤0} (5)；
wherein 
Looking at the definition of γ (i.e., equation (4)), it is known that γ tends to be positive to infinity at time L. The P (gamma) approach to positive infinity; feedback controllers based on the event triggering mechanism of equations (2) and (5) are therefore physically unrealizable. In addition, gamma is in the time zone t epsilon L, + -infinity) is not defined in the text, causing the controller (2) to operate in the time zone t e L, + -infinity) also in are not defined. In order to design a physically realizable controller, a gamma design method is given:
wherein ,γ* Is a constant large enough to be a time L close enough to L * (L * < L) time γ. In addition, when t > L * When due to
Thus P e ＝0,f(t)＝x T P γ* BB T P γ* e and the event trigger mechanism becomes:
t i+1 ＝inf{t＞t i :βγ * x T P γ* x-2e T P γ* bb T P γ* e}；
here P γ* Is when γ=γ in equation (3) * The only positive solution at that time.
Step 3: according to the event trigger mechanism designed in the step 2, calculating to obtain the minimum trigger time interval tau e Is an expression of (2).
Obtaining event trigger time interval function according to the event trigger condition designed in the step 2
The minimum triggering time interval is then the value of the function S from 0 to +.>
The time required.
First deriving the function S
The minimum trigger time interval of the event trigger mechanism designed according to the comparison quotation is as follows:
and then performing simulation verification. In the simulation, the initial state is selected to be x (0) = [1010-1010] T The sampling time is 0.0001s, and the convergence time l=3s is selected. Consider two different cases, case 1: constructing a general designated time controller; case 2: a specified time controller based on event triggering is constructed by step 1 and the selection parameter β=0.1.
TABLE 1
As can be seen from fig. 2 and table 1, under the condition of the same control performance, no matter the minimum trigger time interval or the average trigger time, the control algorithm designed by the present invention is far greater than the general specified time control algorithm, that is, the method designed by the present invention can greatly reduce the update times of the actuator while ensuring the control performance.
Claims (5)
1. The high-precision control method for the designated time based on event triggering is characterized by comprising the following steps of:
step 1: designing a specified time high-precision controller based on a parameter Lyapunov equation:
wherein ,
at time t for system state i The state of the moment, i is N, N isInteger set,/->
Solution P to the parametric Lyapunov equation is at time t i Value of time of day->
Step 2: the following event trigger mechanism based on the parameter Lyapunov equation is designed to determine the update time t of the controller i ：
wherein ,
e is the measurement error, and is the error in the measurement,
wherein the initial value of gamma is +.>
γ * Is at a time L sufficiently close to L * The value of the time gamma, L is any given time, L * < L, beta.E (0, 1) is a designed parameter,
α 2 ＝2dλ max (P n ) N represents the dimension of the system state,
E n ＝diag(n-1,n-2,…,1,0)，P n p (1) is the value of the solution P of the parametric Lyapunov equation when γ=1, λ max Max, diag represent the maximum eigenvalue, maximum and diagonal matrix, c, respectively kj ≥0，k＝1,2,…,n,j＝1,2…,k；
Step 3: according to the event triggering mechanism designed in the step 2, calculating to obtain the following expression of the minimum triggering time interval:
2. the high-precision control method based on the specified time triggered by the event according to claim 1, wherein the specific steps of the step 1 are as follows:
step 11: the nonlinear system model considered is expressed as:
where x is a state variable of the system,
for the derivative of the state variable, u is the control input to the system, and the state matrix for the matrix A and b systems is in the specific form as follows:
Step 12: the method comprises the steps of designing a specified time high-precision controller based on a parameter Lyapunov equation, wherein the specified time high-precision controller is specifically expressed as:
wherein ,
at time t for system state i The state of time, i belongs to N, N is an integer set,/->
The solution P to the parametric Lyapunov equation is at time t as follows i Value of time of day
A T P+PA-Pbb T P＝-γP；
wherein
Gamma is a time-varying parameter, L is any specified time and t.epsilon.0, L).
3. The event trigger based high precision control method for specified time according to claim 1, wherein in the step 2, the measurement error e is defined as follows:
4. the event-triggered based high-precision control method for specified time according to claim 1, wherein in said step 2, when t > L * In the time-course of which the first and second contact surfaces,
the event triggering mechanism becomes:
5. the event trigger-based high-precision control method for the specified time according to claim 1, wherein the specific steps of the step 3 are as follows:
step 31: obtaining an event trigger time interval function according to the event trigger mechanism designed in the step 2
The minimum triggering time interval is the value of the event triggering time interval function S from 0 to +.>
The time required;
step 32: deriving the event trigger time interval function S, and obtaining:
step 33: the minimum trigger time interval of the designed event trigger mechanism is obtained according to the comparison quotation:
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