CN113031444B - Design method of tilting mirror controller based on index optimization - Google Patents
Design method of tilting mirror controller based on index optimization Download PDFInfo
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Abstract
The invention discloses a tilt mirror controller design method based on index optimization. A method for analyzing and designing a tilting mirror controller based on index optimization is provided. The method is different from the PID control and frequency domain correction design method which is widely adopted in the design of the tilting mirror controller in the current photoelectric tracking system. An optimal control theory is introduced to assist in analyzing and designing the tilting mirror controller. A novel design process of the tilting mirror controller based on the index evaluation function is provided. The method provides a theoretical basis for designing the tilting mirror controller based on index optimization, and simplifies the design steps of the tilting mirror controller in the photoelectric tracking system. The controller has the advantages of small overshoot, good rapidity, simple design steps and convenient engineering realization.
Description
Technical Field
The invention belongs to the technical field of tracking control in photoelectric capturing and tracking systems such as photoelectric theodolites, and particularly relates to a design method of a tilting mirror controller based on index optimization.
Background
The tilting mirror has high response speed and high tracking precision, is widely applied to a compound axis tracking system such as a photoelectric telescope, and has more complex motion characteristics, higher speed, acceleration and attitude rate along with the progress of modern science and technology. Meanwhile, as the capturing and tracking system based on the motion platform is used in a large amount, the quick response capability, the target detection capability and the target tracking capability of the system face huge challenges. How to accurately design a tilting mirror controller according to the motion characteristics of a target and different control process requirements is a difficult problem at present.
At present, a tilting mirror controller in a photoelectric tracking system is designed by adopting a traditional PID control strategy, relevant PID parameters are designed by modeling mechanical and electrical characteristics of a tilting mirror, and the tilting mirror control is realized by a method of repeatedly debugging engineering. For the traditional PID control, a large amount of time is needed for repeatedly setting control parameters, and the performance indexes of the controller cannot be intuitively expressed. With the popularization of digital controllers, the problem is well solved by a frequency domain correction strategy, engineers visually analyze and design the controller by using frequency response characteristics of a controlled object, such as cut-off frequency, phase margin, amplitude margin and other frequency domain characteristics, setting steps of controller parameters are simplified, and performance indexes of the controller can be visually expressed.
At present, a design method of a tilting mirror controller based on frequency domain correction can meet application requirements of most occasions, but with more complex target motion characteristics and more complex tracking modes of a photoelectric tracking system, the frequency domain correction method is difficult to accurately design the tilting mirror controller according to the target motion characteristics and different control requirements. Therefore, a design method of a tilting mirror controller capable of adapting to different target motion characteristics according to different control requirements needs to be researched.
Disclosure of Invention
The invention solves the following tracking control problems: the design method of the tilting mirror controller based on index optimization is provided, and the defect that the tilting mirror in the existing photoelectric tracking system cannot design the controller according to different control requirements and target motion characteristics is overcome. The design steps of the tilting mirror controller based on the novel optimal control are simplified.
The technical scheme adopted by the invention is as follows: a design method of a tilting mirror controller based on index optimization is realized by the following steps:
step (1), selecting a proper index evaluation function according to the requirement of a control process, wherein the control requirement of a tilting mirror in the photoelectric tracking system is as follows: the overshoot amount is small while the given signal is quickly responded, the overshoot caused by a larger initial error can be reduced by adopting the integral of time and the error, and the response speed and the overshoot amount of the system are considered in the control process due to the fact that the index evaluation function has time and the error;
the evaluation index function model is as follows:
wherein J is the value of the evaluation function, when J reaches the minimum value, the control system is optimal, t is the control time, t is0Is an initial state time, tfFor the last state time, e (t) is the error between the given angular position and the current angular position of the tilting mirror, u (t) is the unit step input, and similarly, if the control process requirement is that the tracking error is minimum;
the evaluation index function model is as follows:
step (2), establishing a transfer function model G(s) of the tilting mirror
Combining the mechanical characteristic and the electrical characteristic of the tilting mirror, using a second-order link to represent the mechanical characteristic of the tilting mirror, using 2 first-order links to represent the electrical characteristic of the tilting mirror, using an anti-resonance link to represent the high-order resonance characteristic of the tilting mirror,
wherein s is a Laplace operator, K controls gain, and xi is a first-order resonance damping coefficient,
is an intermediate frequency pole, and is,
is a high frequency pole, ωaIs a first order natural frequency, omega, of a tilting mirror1Resonant valley frequency, omega, being the inverse resonant frequency of the tilting mirror2Resonant peak frequency, ξ, being the inverse resonant frequency of the tilting mirror1Damping coefficient, ξ, for the anti-resonance valley2Considering that the higher-order anti-resonance peak of the tilting mirror is small and the resonance frequency point ω is a damping coefficient of the anti-resonance peak1,ω2Much greater than the natural frequency omegaaThe anti-resonance element can be ignored, so the transfer function g(s) of the tilting mirror can be simplified to transfer function (4), and further to transfer function (5):
wherein, bjJ is n-1, … 1 is the controlled object transfer function coefficient, K is the controlled object gain, and s is the laplacian operator;
step (3), establishing a general form of a closed loop transfer function of the tilting mirror:
where Φ(s) is a closed-loop transfer function, C(s) is a controller, G(s) is a controlled object, Y(s) is a system output, U(s) is an input signal, s is a Laplace operator, σ(s) is a controlled object, andjis a closed loop transfer function coefficient, where j is 0,1,2.. n-1,
for closed loop transfer function gain factor, in which
To reduce computational complexity, the closed-loop transfer function (6) is solved to a standard time ω using a normalized differential equationnt, so that the Y-dimensional space described in the transfer function (6), where Y ═ n, can be reduced to a Y-1 dimensional space, for a standard Y-order system Φ(s) only Y-1 parameters a need to be determined1,a2…an-1The value of phi(s) can be determined,
wherein ω isnIs a natural frequency, ajIs a closed loop transfer function coefficient, where j is 1,2 … n-1;
step (4), establishing a functional related to the closed-loop function phi(s) by using the index evaluation function J established in the step (1) and the closed-loop transfer function equation phi(s) established in the step (3), and solving the optimal closed-loop transfer function coefficient a when the index evaluation function is minimumjWhere j is 1,2 … n-1, let expression e be yref-y (t) into functional (1), functional (1) can be reduced to functional (8), where yrefFor a given amount of tilt mirror position, y is the true angular position of the tilt mirror, and e is the error between the given angular position and the true angular position. Considering that the given quantity is typically a step input, the functional (8) can be reduced to a functional (9) where u (t) is the unit step input,
and (5) for a high-order linear system phi(s) exceeding the third order, the geometric meaning that the index evaluation function J takes the minimum value is that the generalized area of the multidimensional phase plane error is the minimum, and the index evaluation function J takes the solution a of the minimum valuejWhere j is 1,2 … n-1, is the minimum point of the multidimensional phase plane, and the dimension of the system is the number of states described by the closed-loop transfer function. Rewriting the closed-loop transfer function phi(s) into a differential equation form, where omega is takenn1, obtaining a time domain expression (10) of an angle error e (t) of the tilting mirror by subtracting the output angular position y (t) from the step input 1(t), obtaining a functional expression (11) of an optimal index by integrating the angle error e (t) with time,
wherein, in the time domain expression (10), y (t) is the true angular position of the tilting mirror,
in order to be the first order differential of the angular position,
second order differential of angular position, and so on yn(t) is the angular position nth order differential, ajN-1 is the optimal closed loop transfer function coefficient, e (t) is the error between a given angular position and the true angular position. In the functional expression (11), J is an index evaluation function.
Step (6) to order aj1, by changing only an-1The parameter a of the index function J can be obtainedn-1Curve J ═ f (a)n-1,an-2＝const,…,a1Const), let ajJ-n-2, n-3, 2,1 takes different given values, i.e. ajThe parameter a of the indicator function J is obtained from const1, const2, …, where J is n-2, n-3, 2,1n-1Continuously change, ajA at different given valuesjHypersurface expression (12) of 2,1, n-2, n-3, …, J-n-1, const2, and J-a are further obtainedn-1-an-2…-a1The same parameter a in n-dimensional spacen-1At different given values, i.e. ajConst1, const2, … where j is n-2, n-3, a hypersurface section at 2,1, the hypersurface shape follows a given value ajJ is different from n-2, n-3 …,1, but the optimum performance parameter an-1,an-2…,a1Finally falling to the bottommost part of the hypersurface to obtain a solved parameter ajJ is n-1, n-2 …, and 1 is the current index evaluation function J (a)n-1,an-2,…,a1) Taking the optimal performance index parameter at the minimum value;
step (7), the closed loop transfer function (7) established in the simultaneous step (3) and the optimal performance parameter a solved in the step (6)jJ is n-1, n-2 …,1, and the optimum closed loop transfer function phi is foundop(s)：
Wherein ω isnNatural frequency, beta, of the transfer function of the closed loop to be optimizedj＝ajWherein j is n-1, n-2, …, 1;
step (8) according to the optimal closed loop transfer function phiop(s) designing a controller c(s) which, considering that the closed loop tracking system is a system without a static error, is designed to:
wherein KiAnd i is 0,1,2 …, and n-2 is a controller parameter to be designed, and is combined with the tilting mirror transfer function G(s) established in the step (2) and the optimal closed-loop transfer function phi established in the step (7)op(s) the controller transfer function C(s) established in step (8), the simultaneous transfer functions (5), (13), (14) solving the current closed loop transfer function phir(s)，
Wherein phir(s) is the solved closed loop transfer function, bjTransferring function parameters for the controlled object, wherein j is n-1, … 1;
step (9) solving the optimal controller parameter KiI-0, 1,2 …, n-2 and the natural frequency ω of the optimal closed-loop transfer functionnComparing the current closed loop transfer phirDenominator of(s) and optimum closed loop transfer function ΦopObtaining an equation set (16) by the denominator of(s), and solving the equation set (16) to obtain the optimal controller parameter KiI-1, 2 …, n-2 and the natural frequency ω of the optimum closed-loop transfer functionn，
Wherein beta isjFor the optimal closed loop transfer function coefficient solved in step (6) where j is n-1, … 1, bjFor the tilted mirror transfer function parameter established in step (2) where j ═ n-1, … 1, K is the tilted mirror gain coefficient established in step (2), K isiI is 1,2 …, and n-2 is the optimal controller parameter to be solved;
step (10) comparing the optimal closed loop function phiop(s) and solved closed loop function phir(s) solving for pre-filter p(s);
utilizing the closed loop transfer function phi obtained in step (8)r(s) the optimal closed-loop transfer function Φ obtained in step (6)op(s) designing a pre-filter P(s) and solving the closed loop transfer function phir(s) correction to the optimum closed loop transfer function, i.e. phiop(s)＝P(s)Φr(s)，
Wherein K is the gain of the controlled object established in the step (2), and KiI is 1,2 …, n-2 is the optimal controller parameter in step 9, parameter K, KiAll obtained in step (2) and step (9), so that the step does not need to redesign the prefilter P(s);
step (11) of setting an optimal closed loop transfer function phiop(s)
Setting standard optimal closed loop transfer function phi of different orders based on different index evaluation function Jop(s) facilitating the use of table lookup in the next tilting mirror controller design, reducing the standard optimalityAnd the closed-loop transfer function parameter setting time simplifies the parameter setting step of the tilting mirror controller based on novel optimal control.
Table 1 shows the standard optimal closed-loop transfer function Φ for different system orders Y of 2,3 … n based on different index evaluation functions Jop(s) the next time the controller of the tilting mirror is designed, the table is looked up and used conveniently.
TABLE 1
The method for selecting the proper evaluation index function and selecting the proper index function J according to different control requirements comprises the following steps: if the control process emphasizes the influence of the recent response and reduces the influence of a larger initial error on the index evaluation function;
then consider the index evaluation function:
if the control process emphasizes that the error is eliminated at the fastest speed, the movement time of the system transferred from the initial state to the final state is shortest;
then consider the index evaluation function:
if the control process emphasizes that the system has the minimum tracking error, the tilting mirror is always kept at the minimum tracking error;
then consider the index evaluation function:
where u (t) is the unit step response, t0Is an initial state time, tfIs in the final stateThe state time, e (t), is the error of the given angular position of the tilting mirror from the current angular position.
The described J-an-1-an-2…-a1The same curved surface is different from a in n-dimensional spacejIn a high-order linear system, the optimal index J represents the minimum generalized area of an error, and the optimal performance index parameter beta of the optimal closed-loop transfer function phi(s) is difficult to obtain intuitively and quickly by adopting an analytic methodjJ is n-1, n-2 …,1, so the optimum performance index parameter beta is obtained by an experimental methodjJ is n-1, n-2 …,1, let ajContinuously varying parameters using mathematical analysis software, const1, j-n-1, n-2 …,1
A series of parameters of the index function J can be obtained
Continuously changing values, connecting the values into a smooth curve by using mathematical drawing software
Taking different given values ajJ-a is further obtained from const1, const2, …, const (n-1), J-n-2, n-3 …,1n-1-an-2…-a1The same parameter a in n-dimensional spacen-1At a different point ajThe optimal performance index parameter of the index function J falls on the bottom of the hypersurface under the hypersurface tangent plane of the 1 value of const1, const2, … and const (n-1), and J is n-2 and n-3 …, and is marked as aj,j＝n-1,n-2…,1。
Compared with the prior art, the invention has the advantages that:
the tilting mirror controller design method based on index optimization is high in response speed and small in overshoot. The designed controller is only lower in order than a controlled object by one order, so that the designed controller can restrain high-order states (state variables such as speed and acceleration) of the system, when control requirements are the same, the optimal closed-loop transfer function set by using the index optimization design method can be repeatedly used, optimal parameter setting time is greatly shortened, the technical requirements of the existing photoelectric tracking system for the rapidness and the stability of the tilting mirror are met, and the controller has strong adaptability. The problem that a tilting mirror in a photoelectric tracking system cannot design a controller according to different control requirements and target motion characteristics is solved. The method simplifies the tilting mirror parameter setting step based on novel optimal control.
Drawings
FIG. 1 is a control block diagram of a tilt mirror controller design method based on index optimization according to the present invention;
FIG. 2 is a frequency domain plot of the tilt mirror;
FIG. 3 is a Simulink global simulation and Simulink detailed simulation block diagram, wherein FIG. 3(a) is a Simulink global simulation block diagram, and FIG. 3(b) is a Simulink detailed simulation block diagram;
FIG. 4 is a graph comparing step responses of conventional PID control, conventional frequency domain correction, and index-based optimization control;
FIG. 5 is a flow chart of a tilt mirror controller design based on index optimization.
Detailed Description
The invention is further described with reference to the following figures and detailed description.
The invention relates to a tilt mirror controller design method based on index optimization, which comprises the following concrete implementation steps:
(1) and selecting a proper evaluation index function according to the requirement of the control process.
In general, the control requirements of a tilting mirror in an electro-optical tracking system are: the overshoot amount is small while the given signal is quickly responded, the overshoot caused by a large initial error can be reduced by adopting the integral of time and the error, and the response speed and the overshoot amount of the system are considered in the control process because the index evaluation function has time and the error.
The evaluation index function model is as follows:
wherein J is the value of the evaluation function, when J reaches the minimum value, the control system is optimal, t is the control time, t is0Is an initial state time, tfAt the end time, e (t) is the error of the given angular position of the tilting mirror from the current angular position, u (t) is the unit step input. Similarly, if the control process requires the minimum tracking error.
The evaluation index function model is as follows:
the specific implementation steps will be described by taking an index evaluation function of integration of time and error as an example.
(2) A transfer function model g(s) of the tilting mirror is established as in fig. 2, as in transfer function (3.1),
the mechanical characteristic and the electrical characteristic of the tilting mirror are combined, a second-order link is used for representing the mechanical characteristic of the tilting mirror, two first-order links are used for representing the electrical characteristic of the tilting mirror, and an anti-resonance link is used for representing the high-order anti-resonance of the tilting mirror. Wherein s is a Laplace operator, K controls gain, and xi is a first-order resonance damping coefficient,
is an intermediate frequency pole, and is,
is a high frequency pole, ωaIs a first order natural frequency, omega1Valley frequency of resonance, omega, being the anti-resonance frequency2Resonance peak frequency, ξ, of the antiresonance frequency1Damping coefficient, ξ, for the anti-resonance valley2The damping coefficient of the anti-resonance peak is shown. Considering that the anti-resonance peak of the tilting mirror is small and the anti-resonance frequency omega1,ω2Much greater than the natural frequency omegaaThe anti-resonance link can be ignored.Therefore, the transfer function G(s) of the controlled object can be simplified into a transfer function (3.2),
wherein b isjJ is a transfer function parameter of the tilting mirror, 1,2,3, 4;
(3) establishing a general form of a closed-loop transfer function
Where Φ(s) is a closed-loop transfer function, C(s) is a controller, G(s) is a controlled object, Y(s) is a system output, U(s) is an input signal, s is a Laplace operator, σ(s) is a controlled object, andjthe closed loop transfer function coefficients, where j is 0,1,2.. n-1,
for closed loop transfer function gain factor, in which
To reduce the computational complexity, the closed-loop transfer function (4) is solved to a standard time ω using a normalized differential equationnAnd (5) standard closed-loop transfer function of order n of t. Thus, the n-dimensional space can be reduced to n-1-dimensional space, and only n-1 parameters a need to be determined for the standard n-order system phi(s)1,a2…an-1The value of phi(s) can be determined,
wherein ω isnIs the natural frequency. a isjFor closed-loop transfer function coefficients, where j is 1,2 … n-1
(4) Combining the index evaluation function J established in the step (1) and the closed-loop transfer function equation phi(s) established in the step (3), establishing a functional related to the closed-loop function, and solving the current index evaluation functionOptimal closed loop transfer function coefficient a at minimum valuejLet expression e be yref-y (t) into functional (1), functional (1) can be reduced to functional (6), where yrefFor a given amount of tilt mirror position, y is the true angular position of the tilt mirror, and e is the error between the given angular position and the true angular position. Considering that the given quantity is typically a step input, the functional (6) can be simplified to a functional (7) where u (t) is the unit step input signal.
(5) For a high-order linear system phi(s) exceeding the third order, the geometric meaning that the index evaluation function J takes the minimum value is that the generalized area of the multidimensional phase plane error is the minimum, and the index evaluation function J takes the solution a of the minimum valuejWhere j is 1,2 … n-1, is the minimum point of the multidimensional phase plane, and the dimension of the system is the number of states described by the closed-loop transfer function. Rewriting closed-loop transfer function phi(s) into time domain expression, wherein omega is takenn1. And (3) subtracting the output angular position y (t) from the unit step input 1(t) to obtain a time domain expression (8) of the angular error e (t) of the tilting mirror. And (4) integrating the angle error e (t) with time to obtain a functional expression (9) of the optimal index.
Wherein, in the time domain expression (8), y (t) is the true angular position of the tilting mirror,
in order to be the first order differential of the angular position,
second order differential of angular position, and so on yn(t) is the angular position nth order differential, ajN-1 is the optimal closed loop transfer function coefficient, e (t) is the error between a given angular position and the true angular position. In functional expression (9), J is an index evaluation function.
(6) Let aj1, by changing only an-1The parameter a of the index function J can be obtainedn-1Curve J ═ f (a)n-1,an-2＝const,…,a1Const), let ajJ-n-2, n-3, 2,1 takes different given values, i.e. ajThe parameter a of the indicator function J is obtained from the const1, const2, … const (n-2), where J is n-2, n-3, 2,1n-1Continuously change, ajA at different given valuesjHypersurface expression (10) of 2,1 ═ const1, const2, …, j ═ n-2, n-3. Further obtaining J-an-1-an-2…-a1The same parameter a in n-dimensional spacen-1At different given values, i.e. ajConst1, const2, … const (n-2), where j is n-2, n-3, a hypersurface section at 2,1, the hypersurface shape follows a given value ajJ is different from n-2, n-3 …,1, but the optimum performance parameter an-1,an-2...,a1Will eventually fall to the very bottom of the hypersurface. Solved parameter ajJ is n-1, n-2 …, and 1 is the current index evaluation function J (a)n-1,an-2,…,a1) Taking the optimal performance index parameter at the minimum value,
(7) utilizing the optimal performance parameter a solved in the step (6)jAnd j is n-1, n-2 …,1, the standard closed-loop transfer function established in the simultaneous step (3), and considering that the tilting mirror object is 4 th order, the optimal closed-loop transfer function is taken as:
wherein ω isnNatural frequency, beta, of the transfer function of the closed loop to be optimizedj＝ajWherein j is 1,2,3, 4.
(8) According to the optimum closed loop transfer function phiop(s) design controller C(s). Considering that the closed loop tracking system is a no-static-error system and the optimal closed loop transfer function phiopThe order of(s) is 5, and the optimal controller C(s) is selected as:
wherein, KiAnd i is 0,1,2 and 3 which are parameters of the controller to be designed. Combining the tilted mirror transfer function G(s) established in the step (2) and the optimal closed-loop transfer function phi established in the step (7)op(s) the controller transfer function C(s) established in step (8), simultaneous expressions (3.2), (11) and (12), solving the current closed loop transfer function phir(s)。
Wherein phir(s) is the solved closed loop transfer function, bjJ is 1,2,3,4 as the controlled object transfer function parameter.
(9) Solving for optimal controller parameter KiI is 0,1,2,3 and the optimum closed loop transfer function natural frequency ωn。
Comparing current closed loop transfer phirDenominator of(s) and optimum closed loop transfer function ΦopThe denominator of(s) yields the system of equations (14).
Solving the equation set (14) can solve the optimal controller parameter KiI-0, 1,2,3 and the natural frequency ω of the optimal closed-loop transfer functionn。
(10) By comparing the optimum closed loop function phiop(s) and closed loop function phir(s) obtaining a prefilter P(s)
Comparing the closed loop transfer function phi obtained in the step (8)r(s), the optimal closed loop transfer function Φ obtained in step 7op(s). Designing a pre-filter P(s) and solving the solved closed loop transfer function phir(s) correction to the optimal closed loop transfer function Φop(s) wherein Φop(s)＝P(s)Φr(s)。
Where K is the tilting mirror gain, KiAnd i is 0,1,2 and 3, which are solved optimal controller parameters. Parameter K0,K1,K2,K3,ωnAll come from the optimal controller c(s) and therefore there is no need to design the pre-filter p(s) again.
(11) Setting an optimal closed loop transfer function phiop(s) is shown in Table 1
Setting standard optimal closed loop transfer function phi of different orders based on different index evaluation function JopAnd(s) table lookup is convenient for the next time of designing the controller of the tilting mirror, the parameter setting time of the standard optimal closed loop transfer function is shortened, and the parameter setting step of the tilting mirror controller based on index optimization is simplified.
TABLE 1
Compared with the prior art, the invention has the advantages that:
(1) compared with the traditional method for setting the parameters of the tilting mirror controller, the method introduces different index evaluation functions aiming at different control requirements and different tracking target characteristics, and utilizes the method for designing the tilting mirror controller based on index optimization, so that an engineer can track the motion state of a target according to the requirements of a control process, and specifically designs the controller of the tilting mirror, thereby overcoming the defect that the traditional tilting mirror controller design cannot adapt to different motion processes and different target characteristics.
(2) The design steps of the tilting mirror controller based on index optimization are simplified, the optimal closed-loop transfer functions of different orders based on different evaluation functions are set by adopting an experimental method, and the parameters are basically unchanged after the optimal closed-loop transfer function parameters are set. The controller can be directly used when being designed next time without setting again. Greatly shortens the design time of the tilting mirror controller and is easier to realize in engineering.
(3) Compared with the traditional tilting mirror controller design, the order of the tilting mirror controller based on index optimization is only one order lower than the transfer function of the tilting mirror object, and the controller can restrain the high-order state of the controlled object, such as the differential of the acceleration, so that more accurate control is realized. The controller has the advantages of high convergence speed, small overshoot and good stability, and meets the design requirements of the current engineering application on the tilting mirror controller.
The invention has not been described in detail and is within the skill of the art.
Claims (3)
1. A design method of a tilting mirror controller based on index optimization is characterized by comprising the following steps: the method comprises the following implementation steps:
step (1), selecting a proper evaluation index function according to the requirement of a control process, wherein the control requirement of a tilting mirror in the photoelectric tracking system is as follows: the overshoot amount is small while the given signal is quickly responded, the overshoot caused by a larger initial error can be reduced by adopting the integral of time and an error, and the response speed and the overshoot amount of the system are considered in the control process due to the fact that the time and the error exist in the evaluation index function;
the evaluation index function model is as follows:
wherein J is the value of the evaluation index function, when J reaches the minimum value, the control system is optimal, t is the control time, t is0Is an initial state time, tfFor the last state time, e (t) is the error between the given angular position and the current angular position of the tilting mirror, u (t) is the unit step input, and similarly, if the control process requirement is that the tracking error is minimum;
the evaluation index function model is as follows:
step (2), establishing a transfer function model G(s) of the tilting mirror
Combining the mechanical characteristic and the electrical characteristic of the tilting mirror, using a second-order link to represent the mechanical characteristic of the tilting mirror, using 2 first-order links to represent the electrical characteristic of the tilting mirror, using an anti-resonance link to represent the high-order resonance characteristic of the tilting mirror,
wherein s is a Laplace operator, K is a control gain, ξ is a first-order resonance damping coefficient,
is an intermediate frequency pole, and is,
is a high frequency pole, ωaIs a first order natural frequency, omega, of a tilting mirror1Resonant valley frequency, omega, being the inverse resonant frequency of the tilting mirror2Resonant peak frequency, ξ, being the inverse resonant frequency of the tilting mirror1Damping coefficient, ξ, for the anti-resonance valley2Is an antiresonance peakConsidering that the higher-order anti-resonance peak of the tilting mirror is small and the resonance frequency point omega is small1,ω2Much greater than the natural frequency omegaaThe anti-resonance element can be ignored, so the transfer function g(s) of the tilting mirror can be simplified to transfer function (4), and further to transfer function (5):
wherein, bjJ is n-1, … 1 is the controlled object transfer function coefficient, s is laplacian operator;
step (3), establishing a general form of a closed loop transfer function of the tilting mirror:
where Φ(s) is the closed loop transfer function, C(s) is the controller, Y(s) is the system output, U(s) is the input signal, s is the Laplace operator, σjIs a closed loop transfer function coefficient, where j is 0,1,2.. n-1,
for closed loop transfer function gain factor, in which
To reduce computational complexity, the closed-loop transfer function (6) is solved to a standard time ω using a normalized differential equationnt, so that the Y-dimensional space described in the transfer function (6), where Y ═ n, can be reduced to a Y-1 dimensional space, for a standard Y-order system Φ(s) only Y-1 parameters a need to be determined1,a2…an-1The value of phi(s) can be determined,
wherein ω isnIs a natural frequency, ajIs a closed loop transfer function coefficient, where j is 1, … n-1;
step (4), establishing a functional related to the closed-loop transfer function phi(s) by using the evaluation index function J established in the step (1) and the closed-loop transfer function phi(s) equation established in the step (3), and solving the optimal closed-loop transfer function coefficient a when the evaluation index function has a minimum valuejLet expression e be yref-y (t) into functional (1), functional (1) can be reduced to functional (8), where yrefGiven the amount of tilt mirror position and y the true angular position of the tilt mirror, considering that the given amount is typically a step input, the functional (8) can be further simplified to functional (9), where u (t) is the unit step input,
and (5) for a high-order linear system exceeding the third order, the geometric meaning that the evaluation index function J takes the minimum value is that the generalized area of the multidimensional phase plane error is the minimum, and the evaluation index function J takes the solution a of the minimum valuejWhere j is 1, … n-1, is the minimum point of the multidimensional phase plane, the dimension of the system is the number of states described by the closed-loop transfer function, the closed-loop transfer function Φ(s) is rewritten as a differential equation, where ω is takenn1, obtaining a time domain expression (10) of an angle error e (t) of the tilting mirror by subtracting the step input 1(t) from y (t), obtaining a functional expression (11) of an optimal index by integrating the angle error e (t) with time,
wherein, in the time domain expression (10), y (t) is the true angular position of the tilting mirror,
in order to be the first order differential of the angular position,
second order differential of angular position, and so on yn(t) is the angular position nth order differential, ajN-1 is the optimal closed loop transfer function coefficient, e (t) is the error between a given angular position and the true angular position;
step (6) to order aj1, by changing only an-1The parameter a of the evaluation index function J can be obtainedn-1Curve J ═ f (a)n-1,an-2＝const,…,a1Const), let ajJ-n-2, n-3, 2,1 takes different given values, i.e. ajThe parameter a of the evaluation index function J can be obtained from const1, const2, …, where J is n-2, n-3, 2,1n-1Continuously change, ajA at different given valuesjHypersurface expression (12) of const1, const2, …, J-n-2, n-3 …,2,1, and further obtaining J-an-1-an-2…-a1The same parameter a in n-dimensional spacen-1At different given values, i.e. ajConst1, const2, …, where j is a hypersurface section at n-2, n-3 …,2,1, the hypersurface shape follows a given value ajDifferent from const, j is n-2, n-3 …,1, but the optimal closed-loop transfer function coefficient an-1,an-2…,a1Finally falling to the bottommost part of the hypersurface to obtain a solved parameter ajJ is n-1, n-2 …,1 is the current evaluation index functionJ(an-1,an-2,…,a1) Taking an optimal closed loop transfer function coefficient at a minimum value;
step (7), the closed loop transfer function (7) established in the step (3) and the optimal closed loop transfer function coefficient a solved in the step (6) are connectedjJ is n-1, n-2 …,1, and the optimum closed loop transfer function phi is foundop(s)：
Wherein ω isnNatural frequency, beta, of the transfer function of the closed loop to be optimizedj＝ajWherein j is n-1, n-2, …, 1;
step (8) according to the optimal closed loop transfer function phiop(s) designing a controller c(s) which, considering that the closed loop tracking system is a system without a static error, is designed to:
wherein KiAnd i is 0,1,2 …, and n-2 is a controller parameter to be designed, and is combined with the tilting mirror transfer function G(s) established in the step (2) and the optimal closed-loop transfer function phi established in the step (7)op(s) the controller transfer function C(s) established in step (8), the simultaneous transfer functions (5), (13), (14) solving the current closed loop transfer function phir(s)，
Wherein phir(s) is the solved closed loop transfer function, bjThe transfer function coefficient of the controlled object is j-n-1, … 1;
step (9) solving the optimal controller parameter KiI-0, 1,2 …, n-2 and the natural frequency ω of the optimal closed-loop transfer functionnComparing the current closed loop transfer function phirDenominator of(s) and optimum closed loop transfer function ΦopObtaining an equation set (16) by the denominator of(s), and solving the equation set (16) to obtain the optimal controller parameter KiI-0, 1,2 …, n-2 and the natural frequency ω of the optimal closed-loop transfer functionn，
Wherein beta isjObtaining the optimal closed-loop transfer function coefficient obtained in step (6), wherein j is n-1, … 1;
step (10) by comparing the optimal closed loop transfer function phiop(s) and solved closed loop transfer function phir(s) solving for pre-filter p(s);
utilizing the closed loop transfer function phi obtained in step (8)r(s) the optimal closed-loop transfer function Φ obtained in step (6)op(s) designing a pre-filter P(s) and solving the closed loop transfer function phir(s) correction to the optimum closed loop transfer function, i.e. phiop(s)＝P(s)Φr(s)，
Parameter K, KiAll obtained in step (2) and step (9), so that the step does not need to redesign the prefilter P(s);
step (11) of setting an optimal closed loop transfer function phiop(s)
Setting different order standard optimal closed loop transfer function phi based on different evaluation index functions Jop(s) the next time the controller of the tilting mirror is designed, the table is looked up and used conveniently.
2. A method as claimed in claim 1The design method of the tilting mirror controller based on index optimization is characterized by comprising the following steps: table 1 shows the standard optimal closed-loop transfer function Φ for different system orders Y of 2,3 … n based on different evaluation index functions Jop(s) for use in looking up a table for the next time of tilt mirror controller design:
TABLE 1
3. The method of claim 1, wherein the controller comprises: selecting a proper evaluation index function, and selecting a proper evaluation index function J according to different control requirements, wherein the realization method comprises the following steps: if the control process emphasizes the influence of the recent response and reduces the influence of a larger initial error on the evaluation index function;
then consider the evaluation index function:
if the control process emphasizes that the error is eliminated at the fastest speed, the movement time of the system transferred from the initial state to the final state is shortest;
then consider the evaluation index function:
if the control process emphasizes that the system has the minimum tracking error, the tilting mirror is always kept at the minimum tracking error;
then consider the evaluation index function:
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