CN103558861A - Dynamic switching method for macro-micro composite motion - Google Patents

Abstract

The invention discloses a dynamic switching method for macro-micro composite motion. In a process that a macro platform approaches to a terminal point and decelerates, a micro platform is started in advance, precision positioning is achieved through the micro platform, and meanwhile the macro platform and the micro platform are also stabilized. Switching amplitude threshold values are determined according to structures and dynamic characteristics of the macro platform and the micro platform, and corresponding switching moments are determined. When the macro platform reaches the switching condition in the deceleration process, motion switching is carried out, motion of the micro platform is started, closed-loop control over the micro platform is achieved through absolute gratings, and finally requirements of the platforms for positional accuracy are met.

Description

A kind of dynamic switching method of grand micro-compound motion

Technical field

The present invention relates to electromechanics and technical field of automation, in particular a kind of dynamic switching method of grand micro-compound motion.

Background technology

Grand micro-composite motion platform comprises grand moving and fine motion two parts, and wherein grand motion is used for realizing high speed and large stroke motion, generally servomotor or linear electric motors, driver and grating, consists of; Micromotion, for the precision positioning of implementation platform, is generally comprised of microstructure, piezoelectric ceramics, piezoelectric ceramic actuator and grating.The final mean annual increment movement of platform is output as the synthetic of two moving displacements, is respectively the high speed and large stroke motion of grand platform and the flexible composition that activates of the piezoelectric ceramics of micromotion platform.

Research about grand micro-compound motion at present also concentrates on the configuration aspects of platform mostly, for the changing method research between two-stage motion, be comparatively short of, what conventionally adopt is static changing method, that is: after finishing and stablize, grand movement travel just starts micromotion, to guarantee the positioning precision of platform.This method can guarantee that microfluidic platform motion is not subject to the impact of grand platform motion, can realize the precision positioning of microfluidic platform, but can not meet the motion requirement of hi-Fix under high frequency start and stop, high-speed motion condition under grand platform motion stabilization condition.Another kind method is to adopt the stroke of micromotion as switching threshold δ, once the stroke of grand motion has reached movement travel L-δ, motion switch.The problem of the method is: do not consider the impact of the vibration of the grand motion process of high speed on micromotion, cannot guarantee the hi-Fix of platform.

Grand micro-composite motion platform is mainly used in realizing the high speed of manufacturing equipment, large stroke, high-precision motion requirement.Research about grand micro-compound motion at present also concentrates on the configuration aspects of platform mostly, for the changing method between two-stage motion, also just adopts simple static switching mode, the locator meams of restarting microfluidic platform after grand platform deceleration is stable.Its problem is: two-stage does not have overlapping, consuming time longer between moving, and cannot realize the motion requirement of high speed, frequent start-stop.Another kind method is to adopt the stroke of micromotion as switching threshold δ, once the stroke of grand motion has reached movement travel L-δ, motion switch.The problem of the method is: do not consider the impact of the vibration of the grand motion process of high speed on micromotion, cannot guarantee the hi-Fix of platform.

Summary of the invention

In order to meet manufacturing equipment, especially the motion requirement of electronics manufacturing equipment to precision positioning under high speed high acceleration condition, solve the contradiction of high speed high acceleration moving and hi-Fix, improve to greatest extent performance index and the production efficiency of manufacturing equipment, the present invention proposes a kind of grand micro-compound motion dynamic switching method that is suitable for high speed, large stroke, high-precision motion demand.The method can be determined the amplitude threshold and the corresponding switching instant that switch based on platform structure and dynamic property, starts the motion of microfluidic platform, the precision positioning of implementation platform within the shortest time in the optimal time of platform retarded motion.

Technical scheme of the present invention is as follows:

A kind of dynamic switching method of grand micro-compound motion, when the vibration amplitude of platform is less than or equal to predetermined amplitude threshold Δ, the switching signal of macro/micromotion feeds back to host computer, by host computer, send motion switching command, start the little stroke motion of microfluidic platform, by absolute grating signal close-loop feedback, realize precision positioning, realize the stable of grand platform simultaneously;

Be t vibration stabilization time of grand platform _map, be t vibration stabilization time of microfluidic platform _mip, a certain moment of microfluidic platform in grand platform retarded motion process starts, and this delay time is t _Δ, corresponding to the vibration amplitude of the grand platform of this time, be Δ, this amplitude is for having carried out normalized amount according to overshoot.According to the impulse response of second-order system, in stabilization process, x ₀(t) needed time when value should meet inequality (1), claim that t is adjustment time, i.e. stabilization time; Inequality is:

|x ₀(t)-x ₀(∞)|≤Δ·x ₀(∞) (t≥t _Δ) (1)

In formula, Δ is given amplitude threshold, i.e. motion overshoot, and after being normalized, its variation range is between 0 and 1:

Impulse response function substitution formula (1) is had:

$\left|\frac{e^{-\mathrm{\ξ}{\mathrm{\ω}}_{n}t}}{\sqrt{1-{\mathrm{\ξ}}^2}}\sin ({\mathrm{\ω}}_{d}t+\arctan \frac{\sqrt{1-{\mathrm{\ξ}}^2}}{\mathrm{\ξ}})\right|\≤\mathrm{\Δ},(t\≥t_{\mathrm{\Δ}})---\left(2\right)$

In formula, ω _dfor having damped frequency, ω _nfor natural frequency, ξ is damping ratio, due to

represented curve is the sinusoidal envelope of amount of decrease, by formula (2) abbreviation, is therefore

$\frac{e^{-\mathrm{\ξ}{\mathrm{\ω}}_{n}t}}{\sqrt{1-{\mathrm{\ξ}}^2}}\≤\mathrm{\Δ},(t\≥t_{\mathrm{\Δ}})---\left(3\right)$

Solution inequality has:

$t_{\mathrm{\Δ}}\≥\frac{1}{{\mathrm{\ξ\ω}}_n}\ln \frac{1}{\mathrm{\Δ}\sqrt{1-{\mathrm{\ξ}}^2}}---\left(4\right)$

For general vibrational system, the stable condition of system damping vibration is Δ _∞=0.02, therefore, for grand platform, getting stabilization time is t _map=t _{map, Δ=0.02}, be t the stabilization time of microfluidic platform _mip=t _{mip, Δ=0.02}, the amplitude condition of switching is Δ _t≤ Δ, when the dynamic response amplitude of grand platform reaches amplitude thresholds Δ, the switching of just moving, starts the motion of micromotion platform; According to amplitude switching condition, grand platform should reach steady state (SS) before micromotion is stable, should meet inequality (5):

t _Δ+t _mip≥t _map (5)

Can be in the hope of the relation of switching time and kinetic parameter in formula (4) substitution formula (5):

$\frac{1}{{\mathrm{\ξ}}_{\mathrm{map}}{\mathrm{\ω}}_{\mathrm{nmap}}}\ln \frac{1}{\mathrm{\Δ}\sqrt{1-{{\mathrm{\ξ}}_{\mathrm{map}}}^2}}+\frac{1}{{\mathrm{\ξ}}_{\mathrm{mip}}{\mathrm{\ω}}_{\mathrm{nmip}}}\ln \frac{1}{{\mathrm{\Δ}}_{\∞}*\sqrt{1-{{\mathrm{\ξ}}_{\mathrm{mip}}}^2}}\≥\frac{1}{{\mathrm{\ξ}}_{\mathrm{map}}{\mathrm{\ω}}_{\mathrm{nmap}}}\ln \frac{1}{{\mathrm{\Δ}}_{\∞}*\sqrt{1-{{\mathrm{\ξ}}_{\mathrm{map}}}^2}}---\left(6\right)$

In formula: ξ _mapfor the damping ratio of grand platform, ξ _mipdamping ratio for microfluidic platform;

ω _nmapfor the natural frequency of grand platform, ω _nmipnatural frequency for microfluidic platform;

Amplitude condition when Δ is grand platform switching, i.e. amplitude threshold;

Δ _∞the amplitude of kinematic system while stablizing, i.e. Δ _∞=0.02;

Because damping ratio in general second order link meets 0 < ξ < 0.7, so formula (6) can be reduced to:

$\frac{-\ln \mathrm{\&Delta;}}{{\mathrm{\&xi;}}_{\mathrm{map}}{\mathrm{\&omega;}}_{\mathrm{nmap}}}+\frac{4}{{\mathrm{\&xi;}}_{\mathrm{mip}}{\mathrm{\&omega;}}_{\mathrm{nmip}}}\&GreaterEqual;\frac{4}{{\mathrm{\&xi;}}_{\mathrm{map}}{\mathrm{\&omega;}}_{\mathrm{nmap}}}---\left(7\right)$

Obtain switching and be respectively amplitude and corresponding switching time:

$\mathrm{\&Delta;}\&le;e^{-|\frac{4}{{\mathrm{\&xi;}}_{\mathrm{map}}{\mathrm{\&omega;}}_{\mathrm{nmap}}}-\frac{4}{{\mathrm{\&xi;}}_{\mathrm{mip}}{\mathrm{\&omega;}}_{\mathrm{nmip}}}|{\mathrm{\&xi;}}_{\mathrm{map}}{\mathrm{\&omega;}}_{\mathrm{nmap}}}---\left(8\right)$

$t_{\mathrm{\&Delta;}}=\frac{-\ln \mathrm{\&Delta;}}{{\mathrm{\&xi;}}_{\mathrm{map}}{\mathrm{\&omega;}}_{\mathrm{nmap}}}---\left(9\right)$

Can calculate amplitude condition Δ and the t of switching _Δ, obtained the maximum critical value and the switching instant that switch, can make two-stage platform within the shortest time, reach stable simultaneously.

The dynamic switching method of a kind of grand micro-compound motion that the present invention proposes.At grand platform, approach the process that terminal slows down, pre-cooling microfluidic platform, realizes precision positioning by microfluidic platform, and grand, microfluidic platform also reaches stable simultaneously.The method is by according to structure and dynamic perfromance grand, microfluidic platform, determine the amplitude threshold of switching, and definite corresponding switching instant, while reaching switching condition in grand platform moderating process, the switching of moving, start the motion of microfluidic platform, by absolute grating, realize the closed-loop control of microfluidic platform, finally reach the positioning accuracy request of platform.

Accompanying drawing explanation

Fig. 1 is the dynamic switching flow figure of macro/micromotion platform;

Fig. 2 is the residual oscillation schematic diagram of macro/micromotion two-stage system;

Fig. 3 is the example explanation of macro/micromotion handoff procedure;

Embodiment

Below in conjunction with specific embodiment, the present invention is described in detail.

Grand micro-composite motion platform comprises grand moving and fine motion two parts, and wherein grand motion is used for realizing high speed and large stroke motion, generally servomotor or linear electric motors, driver and grating, consists of; Micromotion, for the precision positioning of implementation platform, is generally comprised of micrometric displacement structure, piezoelectric ceramics, piezoelectric ceramic actuator and grating.The final mean annual increment movement of platform is output as the synthetic of two moving displacements, is respectively the high speed and large stroke motion of grand platform and the flexible composition that activates of the piezoelectric ceramics of micromotion platform.

Structural design based on macro/micromotion platform and parameter setting, set up kinetic model grand, micromotion platform, analyzes dynamic response and the dynamic perfromance of two platforms, determines the correlation parameter of grand platform, microfluidic platform, as damping ratio, natural frequency etc.

At grand motion stage, displacement output end is mainly subject to the impact of grand platform motion, the moment particularly stopping in grand motion, grand platform stop step response as the input signal of microfluidic platform, displacement output end is at a second order vibrational state, and this vibrates stabilization time is sum stabilization time of two platform models.When the vibration amplitude of grand platform meets a certain amplitude condition, start micromotion platform, by the positional information of absolute grating, feed back, realize closed-loop control and the precision positioning of microfluidic platform motion, and in this course, grand platform is tended towards stability gradually.

As shown in Figure 1, its workflow is: the host computer of macro/micromotion platform sends grand movement instruction, platform is driven and to be realized large stroke motion by motor, and the dynamic response that the kinetic model of system based on grand platform and grating feedback signal can real-time analysis platforms obtains the vibration signal of platform; When platform approaches stroke end and slows down, can produce larger vibration, its vibration amplitude can reduce gradually, and needs the regular hour to tend towards stability; Vibration amplitude Δ when platform _twhile being less than or equal to predetermined amplitude threshold Δ, the switching signal of macro/micromotion feeds back to host computer, by host computer, sends motion switching command, starts the little stroke motion of microfluidic platform, by absolute grating signal close-loop feedback, realize precision positioning, realize the stable of grand platform simultaneously.

The mechanical residual oscillation of macro/micromotion two-stage system is usually expressed as the form of Second-order Damped concussion, and its schematic diagram as shown in Figure 2.Be t vibration stabilization time of grand platform _map, be t vibration stabilization time of microfluidic platform _mip, a certain moment of microfluidic platform in grand platform retarded motion process starts, and this delay time is t _Δ, corresponding to the vibration amplitude of the grand platform of this time, be Δ, this amplitude is for having carried out normalized amount according to overshoot.According to the impulse response of second-order system, in stabilization process, x ₀(t) needed time when value should meet inequality (1), claim that t is adjustment time, i.e. stabilization time.Inequality is:

|x ₀(t)-x ₀(∞)|≤Δ·x ₀(∞) (t≥t _Δ) (1)

In formula, Δ is given amplitude threshold, i.e. motion overshoot, and after being normalized, its variation range is between 0 and 1.

Impulse response function substitution formula (1) is had:

$\left|\frac{e^{-\mathrm{\&xi;}{\mathrm{\&omega;}}_{n}t}}{\sqrt{1-{\mathrm{\&xi;}}^2}}\sin ({\mathrm{\&omega;}}_{d}t+\arctan \frac{\sqrt{1-{\mathrm{\&xi;}}^2}}{\mathrm{\&xi;}})\right|\&le;\mathrm{\&Delta;},(t\&GreaterEqual;t_{\mathrm{\&Delta;}})---\left(2\right)$

In formula, ω _dfor having damped frequency, ω _nfor natural frequency, ξ is damping ratio, due to

represented curve is the sinusoidal envelope of amount of decrease, by formula (2) abbreviation, is therefore

$\frac{e^{-\mathrm{\&xi;}{\mathrm{\&omega;}}_{n}t}}{\sqrt{1-{\mathrm{\&xi;}}^2}}\&le;\mathrm{\&Delta;},(t\&GreaterEqual;t_{\mathrm{\&Delta;}})---\left(3\right)$

Solution inequality has:

$t_{\mathrm{\&Delta;}}\&GreaterEqual;\frac{1}{{\mathrm{\&xi;\&omega;}}_n}\ln \frac{1}{\mathrm{\&Delta;}\sqrt{1-{\mathrm{\&xi;}}^2}}---\left(4\right)$

For general vibrational system, the stable condition of system damping vibration is Δ _∞=0.02, therefore, for grand platform, getting stabilization time is t _map=t _{map, Δ=0.02}, be t the stabilization time of microfluidic platform _mip=t _{mip, Δ=0.02}, the amplitude condition of switching is Δ _t≤ Δ, when the dynamic response amplitude of grand platform reaches amplitude thresholds Δ, the switching of just moving, starts the motion of micromotion platform.According to amplitude switching condition, grand platform should reach steady state (SS) before micromotion is stable, should meet inequality (5):

t _Δ+t _mip≥t _map (5)

Can be in the hope of the relation of switching time and kinetic parameter in formula (4) substitution formula (5):

$\frac{1}{{\mathrm{\&xi;}}_{\mathrm{map}}{\mathrm{\&omega;}}_{\mathrm{nmap}}}\ln \frac{1}{\mathrm{\&Delta;}\sqrt{1-{{\mathrm{\&xi;}}_{\mathrm{map}}}^2}}+\frac{1}{{\mathrm{\&xi;}}_{\mathrm{mip}}{\mathrm{\&omega;}}_{\mathrm{nmip}}}\ln \frac{1}{{\mathrm{\&Delta;}}_{\&infin;}*\sqrt{1-{{\mathrm{\&xi;}}_{\mathrm{mip}}}^2}}\&GreaterEqual;\frac{1}{{\mathrm{\&xi;}}_{\mathrm{map}}{\mathrm{\&omega;}}_{\mathrm{nmap}}}\ln \frac{1}{{\mathrm{\&Delta;}}_{\&infin;}*\sqrt{1-{{\mathrm{\&xi;}}_{\mathrm{map}}}^2}}---\left(6\right)$

In formula: ξ _mapfor the damping ratio of grand platform, ξ _mipdamping ratio for microfluidic platform;

ω _nmapfor the natural frequency of grand platform, ω _nmipnatural frequency for microfluidic platform;

Amplitude condition when Δ is grand platform switching, i.e. amplitude threshold.

Δ _∞the amplitude of kinematic system while stablizing, i.e. Δ _∞=0.02.

Because damping ratio in general second order link meets 0 < ξ < 0.7, so formula (6) can be reduced to

$\frac{-\ln \mathrm{\&Delta;}}{{\mathrm{\&xi;}}_{\mathrm{map}}{\mathrm{\&omega;}}_{\mathrm{nmap}}}+\frac{4}{{\mathrm{\&xi;}}_{\mathrm{mip}}{\mathrm{\&omega;}}_{\mathrm{nmip}}}\&GreaterEqual;\frac{4}{{\mathrm{\&xi;}}_{\mathrm{map}}{\mathrm{\&omega;}}_{\mathrm{nmap}}}---\left(7\right)$

Obtain switching and be respectively amplitude and corresponding switching time:

$\mathrm{\&Delta;}\&le;e^{-|\frac{4}{{\mathrm{\&xi;}}_{\mathrm{map}}{\mathrm{\&omega;}}_{\mathrm{nmap}}}-\frac{4}{{\mathrm{\&xi;}}_{\mathrm{mip}}{\mathrm{\&omega;}}_{\mathrm{nmip}}}|{\mathrm{\&xi;}}_{\mathrm{map}}{\mathrm{\&omega;}}_{\mathrm{nmap}}}---\left(8\right)$

$t_{\mathrm{\&Delta;}}=\frac{-\ln \mathrm{\&Delta;}}{{\mathrm{\&xi;}}_{\mathrm{map}}{\mathrm{\&omega;}}_{\mathrm{nmap}}}---\left(9\right)$

According to every characteristic of grand micro-two-stage platform, as damping ratio, natural frequency etc., can calculate amplitude condition Δ and the t of switching _Δ, obtained the maximum critical value and the switching instant that switch, can make two-stage platform within the shortest time, reach stable simultaneously.If get Δ=0.02 even more hour, be while waiting until grand platform complete stability, just to start microfluidic platform, will affect the overall operation time of platform.

Embodiment:

By Dynamic Modeling and the experimental modal of macro/micromotion platform, can be in the hope of: the damping ratio ξ of grand platform _mapbe 0.1571, natural frequency ω _nmapfor 443.0025Hz, the damping ratio ξ of microfluidic platform _mip=0.2368, natural frequency ω wherein _nmip=7815.5614Hz.Substitution formula (8), can obtain its amplitude threshold: Δ≤0.0558.Now, if get Δ=0.05, can calculate: be t grand micro-switching time _Δ=43.04ms, two platforms reach about 50ms stabilization time.Figure 3 shows that the switching schematic diagram of this example.When the amplitude threshold of grand platform motion is defined as being less than or equal to 0.0558 as calculated, in figure, getting its value is 0.05, can obtain thus 43.04ms switching time of microfluidic platform, when the time point 43.04ms of grand platform motion, start microfluidic platform motion, finally can make grand micro-two-stage motion when 50ms time point, reach steady state (SS).

Should be understood that, for those of ordinary skills, can be improved according to the above description or convert, and all these improvement and conversion all should belong to the protection domain of claims of the present invention.

Claims (1)

1. the dynamic switching method of a grand micro-compound motion, it is characterized in that, when the vibration amplitude of platform is less than or equal to predetermined amplitude threshold Δ, the switching signal of macro/micromotion feeds back to host computer, by host computer, send motion switching command, start the little stroke motion of microfluidic platform, by absolute grating signal close-loop feedback, realize precision positioning, realize the stable of grand platform simultaneously;

Be t vibration stabilization time of grand platform _map, be t vibration stabilization time of microfluidic platform _mip, a certain moment of microfluidic platform in grand platform retarded motion process starts, and this delay time is t _Δ, corresponding to the vibration amplitude of the grand platform of this time, be Δ, this amplitude is for having carried out normalized amount according to overshoot.According to the impulse response of second-order system, in stabilization process, x ₀(t) needed time when value should meet inequality (1), claim that t is adjustment time, i.e. stabilization time; Inequality is:

|x ₀(t)-x ₀(∞)|≤Δ·x ₀(∞) (t≥t _Δ) (1)

In formula, Δ is given amplitude threshold, i.e. motion overshoot, and after being normalized, its variation range is between 0 and 1;

Impulse response function substitution formula (1) is had:

$\left|\frac{e^{-\mathrm{\&xi;}{\mathrm{\&omega;}}_{n}t}}{\sqrt{1-{\mathrm{\&xi;}}^2}}\sin ({\mathrm{\&omega;}}_{d}t+\arctan \frac{\sqrt{1-{\mathrm{\&xi;}}^2}}{\mathrm{\&xi;}})\right|\&le;\mathrm{\&Delta;},(t\&GreaterEqual;t_{\mathrm{\&Delta;}})---\left(2\right)$

In formula, ω _dfor having damped frequency, ω _nfor natural frequency, ξ is damping ratio, due to

represented curve is the sinusoidal envelope of amount of decrease, by formula (2) abbreviation, is therefore

$\frac{e^{-\mathrm{\&xi;}{\mathrm{\&omega;}}_{n}t}}{\sqrt{1-{\mathrm{\&xi;}}^2}}\&le;\mathrm{\&Delta;},(t\&GreaterEqual;t_{\mathrm{\&Delta;}})---\left(3\right)$

Solution inequality has:

$t_{\mathrm{\&Delta;}}\&GreaterEqual;\frac{1}{{\mathrm{\&xi;\&omega;}}_n}\ln \frac{1}{\mathrm{\&Delta;}\sqrt{1-{\mathrm{\&xi;}}^2}}---\left(4\right)$

For general vibrational system, the stable condition of system damping vibration is Δ _∞=0.02, therefore, for grand platform, getting stabilization time is t _map=t _{map, Δ=0.02}, be t the stabilization time of microfluidic platform _mip=t _{mip, Δ=0.02}, the amplitude condition of switching is Δ _t≤ Δ, when the dynamic response amplitude of grand platform reaches amplitude thresholds Δ, the switching of just moving, starts the motion of micromotion platform; According to amplitude switching condition, grand platform should reach steady state (SS) before micromotion is stable, should meet inequality (5):

t _Δ+t _mip≥t _map (5)

Can be in the hope of the relation of switching time and kinetic parameter in formula (4) substitution formula (5):

$\frac{1}{{\mathrm{\&xi;}}_{\mathrm{map}}{\mathrm{\&omega;}}_{\mathrm{nmap}}}\ln \frac{1}{\mathrm{\&Delta;}\sqrt{1-{{\mathrm{\&xi;}}_{\mathrm{map}}}^2}}+\frac{1}{{\mathrm{\&xi;}}_{\mathrm{mip}}{\mathrm{\&omega;}}_{\mathrm{nmip}}}\ln \frac{1}{{\mathrm{\&Delta;}}_{\&infin;}*\sqrt{1-{{\mathrm{\&xi;}}_{\mathrm{mip}}}^2}}\&GreaterEqual;\frac{1}{{\mathrm{\&xi;}}_{\mathrm{map}}{\mathrm{\&omega;}}_{\mathrm{nmap}}}\ln \frac{1}{{\mathrm{\&Delta;}}_{\&infin;}*\sqrt{1-{{\mathrm{\&xi;}}_{\mathrm{map}}}^2}}---\left(6\right)$

In formula: ξ _mapfor the damping ratio of grand platform, ξ _mipdamping ratio for microfluidic platform;

ω _nmapfor the natural frequency of grand platform, ω _nmipnatural frequency for microfluidic platform;

Amplitude condition when Δ is grand platform switching, i.e. amplitude threshold;

Δ _∞the amplitude of kinematic system while stablizing, i.e. Δ _∞=0.02;

Because damping ratio in general second order link meets 0 < ξ < 0.7, so formula (6) can be reduced to:

$\frac{-\ln \mathrm{\&Delta;}}{{\mathrm{\&xi;}}_{\mathrm{map}}{\mathrm{\&omega;}}_{\mathrm{nmap}}}+\frac{4}{{\mathrm{\&xi;}}_{\mathrm{mip}}{\mathrm{\&omega;}}_{\mathrm{nmip}}}\&GreaterEqual;\frac{4}{{\mathrm{\&xi;}}_{\mathrm{map}}{\mathrm{\&omega;}}_{\mathrm{nmap}}}---\left(7\right)$

Obtain switching and be respectively amplitude and corresponding switching time:

$\mathrm{\&Delta;}\&le;e^{-|\frac{4}{{\mathrm{\&xi;}}_{\mathrm{map}}{\mathrm{\&omega;}}_{\mathrm{nmap}}}-\frac{4}{{\mathrm{\&xi;}}_{\mathrm{mip}}{\mathrm{\&omega;}}_{\mathrm{nmip}}}|{\mathrm{\&xi;}}_{\mathrm{map}}{\mathrm{\&omega;}}_{\mathrm{nmap}}}---\left(8\right)$

$t_{\mathrm{\&Delta;}}=\frac{-\ln \mathrm{\&Delta;}}{{\mathrm{\&xi;}}_{\mathrm{map}}{\mathrm{\&omega;}}_{\mathrm{nmap}}}---\left(9\right)$

Can calculate amplitude condition Δ and the t of switching _Δ, obtained the maximum critical value and the switching instant that switch, can make two-stage platform within the shortest time, reach stable simultaneously.

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