CN112684710B - Light beam jitter suppression method based on LQG + PI mixed control strategy - Google Patents
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Abstract
The invention discloses a light beam jitter suppression method based on an LQG + PI mixed control strategy, which comprises the following steps: firstly, establishing a jitter model for a light beam jitter signal, analyzing a corresponding PSD curve of the light beam jitter signal, and performing least square fitting of actually measured data and calculated data to extract and obtain frequency components and obtain accurate description of the light beam jitter signal; secondly, constructing a state vector space based on the light beam jitter signal, establishing LQG control based on a Kalman Filter (Kalman Filter), accurately predicting the jitter signal, and calculating by using the predicted jitter signal to obtain a corresponding control voltage; and then, the voltage is superposed with the control voltage calculated by the parallel PI controller to obtain the final control voltage, thereby completing the real-time correction of the wavefront distortion. The most obvious advantages of the method are that the LQG control can well inhibit the dithering of the high-frequency narrow-band light beam and the PI control has good inhibition capability on the dithering of the low-frequency wide-band light beam, the correction residual error is greatly reduced, the control bandwidth is improved, and the real-time performance is strong.
Description
Technical Field
The invention belongs to the technical field of wave-front processing control, and relates to a light beam jitter suppression method based on an LQG + PI mixed control strategy, which is suitable for wave pre-processing of an adaptive optical system.
Background
In the process of light beam propagation, due to interference of external factors, the optical axis of the light beam can deflect, and then the propagation direction of the light beam can change continuously. The phenomenon is called light beam jitter and mainly comprises low-frequency broadband disturbance caused by transmission media such as atmospheric turbulence and high-frequency narrowband disturbance caused by vibration of mechanical devices, light beam platforms and the like. Seeking a light beam vibration suppression method for effectively suppressing low-frequency disturbance and high-frequency vibration is a direction which is continuously explored by scholars at home and abroad. The traditional PI control method is simple in design, small in calculation amount, widely used in engineering and good in suppression effect on low-frequency broadband disturbance. However, most parameter setting depends on manual experience, and the parameter setting does not have the prediction capability on time-varying disturbance and the control capability on narrow-band disturbance is insufficient. Especially in a complex motion environment, simple PI control is difficult to suppress the light beam jitter of hundreds of hertz, and even amplifies the high-frequency light beam jitter outside the control bandwidth. The LQG control is an optimal control technology based on a state space model, has the substantial advantages of phase prediction capability and full utilization of state space optimal estimation, and has good inhibition capability on high-frequency narrow-band light beam jitter. In 1993, the passhall & Anderson rate first realized LQG-based controllers in AO systems, followed by loze et al, b.le Roux et al, and the kalman filter-based LQG control has received increasing attention. The effectiveness of the LQG control for jitter control was first demonstrated in 2008 by a laboratory. In 2012, Carlos coreia analyzed the discrete time LQG equation equivalent to Minimum Variance (MV) interference rejection. A model identification technology system is successfully implemented on the CANARY, the technical feasibility of spectrum identification is verified, but the method is easy to fall into a local minimum. Therefore, in order to solve the problems of narrow-band high-frequency vibration peaks and wide-band low-frequency disturbance, the invention provides a light beam jitter suppression method based on an LQG + PI mixed control strategy.
Disclosure of Invention
The invention solves the technical problems that: aiming at the problems of low-frequency broadband light beam jitter caused by atmospheric turbulence and high-frequency narrow-band light beam jitter caused by mechanical vibration in an adaptive optical system, a light beam jitter suppression method based on an LQG + PI mixed control strategy is provided.
The technical scheme of the invention is as follows: a beam jitter suppression method based on an LQG + PI mixed control strategy is based on an adaptive optics system control structure, and beam jitter control is realized through the following steps:
step 1: the wavefront sensor obtains light beam jitter signal data containing noise by detecting the offset of the centroid of the light spot;
step 2: establishing a system model for the light beam jitter signal, and expressing the system model by a second-order damped oscillation equation as follows:
wherein
Is a signal that is a dither signal and,
is the first and second time derivative of the jitter signal, K is the damping coefficient, the size is related to the jitter bandwidth, it represents the width and overshoot of the oscillation peak, G is the static gain, xi is the variance of sigma vib Of the oscillating source function, ω 0 Representing the natural oscillation angular frequency, with the magnitude:
ω 0 ＝2πf vib (2)
wherein, f vib Is the dither frequency of the optical beam dither signal.
And step 3: the discrete expression in equation (1) is:
wherein the content of the first and second substances,
representing the beam wobble signal xi at n instants n Zero mean white Gaussian noise, a, for n instants 1 、a 2 Two parameters representing the discrete model, respectively:
the equation (3) is actually a Second-order Auto-Regressive model (AR 2 model), and a discrete model of the beam wobble signal having an actual physical meaning is established.
And 4, step 4: calculating the Power spectral density of the optical beam jitter signal by using a PSD (Power spectral Density) calculation formula of an AR2 model, and using S (f) i ) Expressed as:
wherein σ 2 Power as a function of forced vibration source, a 1 、a 2 Is the model parameter mentioned in formula (4) and formula (5), and j is an imaginary unit.
And 5: calculating the estimated PSD curve of the detected light beam jitter data and the PSD calculated by the formula (6) in the step 4 by utilizing nonlinear least square fitting to identify model parameters to obtain a 1 、a 2 And σ 2 An accurate description of the beam dither signal is established.
Step 6: first, a beam-shake control model based on LQG control is established. Let n time state vector X n Expressed as:
where m represents the mth wobble signal, n represents the nth time,
is the m-th beam dither signal at time n, and T represents the transpose of the matrix.
And 7: according to the beam signal model in steps 3 and 4, n +1State vector X of time n+1 Comprises the following steps:
X n+1 ＝AX n +V n (8)
wherein, a is a coefficient matrix, which is composed of model coefficients of each single-frequency beam wobble signal, and can be identified in step 5, and is specifically represented as:
V n representing the model error, with power σ 2 Is dependent on the oscillator source function xi, m represents the mth dither signal,
the model parameters of the mth jitter signal are the same as those of the formula (4) and the formula (5).
And step 8: assuming that the system is a two frame delay system, the centroid offset y n Expressed as:
y n ＝CX n -DNu n-2 +ω n (10)
where D is the response matrix of the wavefront sensor, N is the response function of the tilting mirror, u n-2 Is the control voltage at time n-2, omega n Is the detection noise of the wave-front detector, the detection noise is the mean value of zero and the variance of sigma ω C is a measurement matrix, and:
C＝D(0,1,0,1,...0,1) (11)
and step 9: by measurement of the signal y n Correcting the predicted value of the model, and carrying out optimal estimation, wherein the iterative process is as follows:
in the formula (12), the reaction mixture is,
represents n time pairs X n Is estimated by the estimation of (a) a,
is n-1 time pair X n Estimation of (H) ∞ Represents the asymptotic gain matrix of the Kalman filter, and the expression is as follows:
H ∞ ＝Σ ∞ C T (CΣ ∞ C T +Σ ω ) -1 (14)
wherein, sigma ω Representing probe noise, ∑ ∞ The covariance matrix representing the model error is an asymptotic solution of the geometric Riccati equation, and the expression is as follows:
Σ ∞ ＝AΣ ∞ A T +Σ v -AΣ ∞ C T (CΣ ∞ C T +Σ ω ) -1 CΣ ∞ A T (15)
wherein, sigma v Is a representative model error V n Is expressed as:
step 10: optimal predicted state for time n to time n +1 after iteration through equations (12) -13
Thereby obtaining an optimal control voltage u n The expression is:
where P denotes an extraction matrix for extracting the disturbance phase from the predicted state vector and obtaining the control voltage by projection to the TTM, expressed as:
P＝N -1 (1,0,10,...,...) (18)
thus, a control voltage u based on the LQG control model is obtained LQG 。
Step 11: then, a PI control model is established, a simplest integral controller is adopted, and a time domain expression of the PI control model is as follows:
u n ＝a*u n-1 +b*u e (19)
wherein u is e Is an error control signal, expressed as:
u e ＝u n -u n-1 (20)
obtaining PI control voltage u PI 。
Step 12: the PI control and the LQG control are connected in parallel to obtain a final control voltage, which is expressed as:
u o ＝u LQG +u PI (21)
step 13: final control voltage u o And the light beam is sent to the tilting mirror, and the tilting mirror is driven to generate corresponding mirror surface deflection, so that the light beam jitter signal is corrected.
Further, fig. 3 is a graph showing the effect of beam shake correction by the system using the method of the present invention, using a conventional PI controller, and using LQG control in an embodiment example. In fig. 3(a), the thick solid line represents the root mean square value of the initial phase, the dash-dot line represents the root mean square value of the residual phase after PI control, the thin solid line represents the root mean square value of the residual phase after LQG control, and the dotted line represents the root mean square value of the residual phase after LQG + PI control. As can be seen from fig. 3(a), the method of the present invention is lower than PI control, LQG control in the correction residual of the dither signal. In fig. 3(b), a thick solid line represents a power spectral density curve of an initial jitter signal, a dashed dotted line represents a power spectral density curve of a jitter signal after PI control, a thin solid line represents a power spectral density curve of a jitter signal after LQG control, and a dotted line represents a power spectral density curve of a jitter signal after LQG + PI control according to the present invention. As can be seen from fig. 3(b), the method of the present invention has significant suppression effect on both low-frequency and high-frequency jitter signals.
Further, compared with the prior art, the invention has the following advantages: the method solves the problem of low-frequency broadband light beam jitter caused by atmospheric turbulence by using good adaptability of PI control to low-frequency broadband disturbance, solves the problem of high-frequency narrowband light beam jitter caused by mechanical vibration by using excellent inhibition capability of LQG control to high-frequency narrowband vibration, greatly reduces wavefront residual error, and effectively improves system control bandwidth.
Drawings
FIG. 1 is a schematic diagram of a closed-loop control architecture for an adaptive optical system;
FIG. 2 is a schematic diagram of a method for suppressing beam jitter based on LQG + PI hybrid control strategy;
fig. 3 is a diagram showing the effect of beam jitter correction using the method of the present invention, using a conventional PI controller, and using LQG control in an exemplary system, where fig. 3(a) is a diagram showing that the correction residual error of the jitter signal in the method of the present invention is lower than that in PI control and LQG control, and fig. 3(b) is a diagram showing that the method of the present invention has significant suppression effect on both low-frequency and high-frequency jitter signals.
Detailed Description
In order to make the technical solutions and advantages of the present invention more apparent, the present invention is further described in detail below with reference to the accompanying drawings in conjunction with specific embodiments.
Fig. 1 is a schematic diagram of a closed-loop Control structure of an adaptive optical system, which is composed of a WFS (wave front Sensor), a CC (Control Cell), a D/a converter (Digital/Analog), and a TTM (Tip-tilt Mirror). The beam-jitter control system is actually a closed-loop control implemented by WFS and TTM. Optical beam dither signal
Compensation correction via TTM
Obtaining a residual signal
The wave front detector detects the residual signal and sends the residual signal to the control unit, and the obtained control voltage u is applied to the TTM to generate surface shape change, thereby obtaining phase correction
And further realize the closed-loop control of the light beam dithering.
Fig. 2 is a schematic diagram of a beam jitter suppression method based on LQG + PI hybrid control strategy, and the adaptive optical system closed-loop control structure shown in fig. 1 further clarifies the structure proposed by the method: and the LQG and the PI controller are connected in parallel, the control voltages obtained by respective calculation are added to obtain the final control voltage, and the TTM is driven to generate the deformation compensation quantity.
Fig. 3 is a graph showing the effect of beam-shake correction using the method of the present invention, using a conventional PI controller, and using LQG control in a system in an embodiment example. In fig. 3(a), the thick solid line represents the root mean square value of the initial phase, the dash-dot line represents the root mean square value of the residual phase after PI control, the thin solid line represents the root mean square value of the residual phase after LQG control, and the dotted line represents the root mean square value of the residual phase after LQG + PI control. As can be seen from fig. 3(a), the method of the present invention is lower than PI control, LQG control in the correction residual of the dither signal. In fig. 3(b), a thick solid line represents a power spectral density curve of an initial jitter signal, a dashed dotted line represents a power spectral density curve of a jitter signal after PI control, a thin solid line represents a power spectral density curve of a jitter signal after LQG control, and a dotted line represents a power spectral density curve of a jitter signal after LQG + PI control according to the present invention. As can be seen from fig. 3(b), the method of the present invention has significant suppression effect on both low-frequency and high-frequency jitter signals. Wherein LQG refers to Linear Quadratic Gaussian (Linear Quadratic Gaussian), PI refers to Proportional-Integral (Proportional-Integral), and PSD refers to Power spectral Density (Power spectral Density).
The present invention is not limited to the specific embodiments described above, which are intended to be illustrative only and not limiting. Those skilled in the art, having the benefit of this disclosure, may effect numerous modifications thereto without departing from the scope and spirit of the invention as set forth in the claims that follow. The invention has not been described in detail and is part of the common general knowledge of a person skilled in the art.
Claims (3)
1. A beam jitter suppression method based on an LQG + PI mixed control strategy is based on an adaptive optics system control structure and is characterized in that beam jitter control is realized through the following steps:
step 1: the wavefront sensor obtains light beam jitter signal data containing noise by detecting the offset of the centroid of the light spot;
step 2: establishing a system model for the light beam jitter signal, and expressing the system model by a second-order damped oscillation equation as follows:
wherein
Is a signal that is a dither signal and,
is the first and second time derivative of the dither signal, K is damping coefficient, the size is related to the dither bandwidth, representing the width and overshoot of the dither peak, G is static gain, xi is variance σ vib Of the oscillating source function, ω 0 Representing the natural oscillation angular frequency, with the magnitude:
ω 0 ＝2πf vib (2)
wherein f is vib Is the dithering frequency of the optical beam dithering signal;
and step 3: the discrete expression in equation (1) is:
wherein the content of the first and second substances,
representing the beam wobble signal xi at n instants n Zero mean white Gaussian noise at n instants 1 、a 2 Two parameters representing the discrete model, respectively:
the formula (3) is actually a Second-order Auto-Regressive model (AR 2 model), and a discrete model of the light beam jitter signal with actual physical significance is established;
and 4, step 4: calculating the Power spectral density of the optical beam jitter signal by using a PSD (Power spectral Density) calculation formula of an AR2 model, and using S (f) i ) Expressed as:
wherein σ 2 Power as a function of forced vibration source, a 1 、a 2 Is the model parameter mentioned in formula (4) and formula (5), and j is an imaginary unit;
and 5: calculating the estimated PSD curve of the detected light beam jitter data and the PSD calculated by the formula (6) in the step 4 by utilizing nonlinear least square fitting to identify model parameters to obtain a 1 、a 2 And σ 2 Establishing accurate description of the light beam jitter signal;
step 6: firstly, a light beam jitter control model based on LQG control is established, and n time is orderedState vector X n Expressed as:
where m represents the mth wobble signal, n represents the nth time,
is the mth beam dither signal at n moments, and T represents the transposition of the matrix;
and 7: according to the light beam signal model in the steps 3 and 4, the state vector X at the moment n +1 n+1 Comprises the following steps:
X n+1 ＝AX n +V n (8)
wherein, a is a coefficient matrix, which is composed of model coefficients of each single-frequency beam wobble signal, and can be identified in step 5, and is specifically represented as:
V n representing the model error, with power σ 2 Is dependent on the oscillating source function xi, m represents the mth dither signal,
the model parameters of the mth jitter signal have the same meanings as formula (4) and formula (5);
and step 8: assuming that the system is a two frame delay system, the centroid offset y n Expressed as:
y n ＝CX n -DNu n-2 +ω n (10)
where D is the response matrix of the wavefront sensor and N is the response function of the tilting mirror，u n-2 Is the control voltage at time n-2, ω n Is the detection noise of the wave-front detector, the detection noise is the mean value of zero and the variance of sigma ω C is a measurement matrix, and:
C＝D(0,1,0,1,...0,1) (11)
and step 9: by measurement of the signal y n Correcting the predicted value of the model, and carrying out optimal estimation, wherein the iterative process is as follows:
in the formula (12), the reaction mixture is,
represents n time pairs X n Is estimated by the estimation of (a) a,
is n-1 time pair X n Estimation of (H) ∞ Represents the asymptotic gain matrix of the Kalman filter, and the expression is as follows:
H ∞ ＝Σ ∞ C T (CΣ ∞ C T +Σ ω ) -1 (14)
wherein, sigma ω Representing probe noise, ∑ ∞ The covariance matrix representing the model error is an asymptotic solution of the geometric Riccati equation, and the expression is as follows:
Σ ∞ ＝AΣ ∞ A T +Σ v -AΣ ∞ C T (CΣ ∞ C T +Σ ω ) -1 CΣ ∞ A T (15)
wherein, sigma v Is a representative model error V n Is represented as:
step 10: after the iteration of formula (12) to formula (13), the optimal prediction state at the time n to the time n +1 can be obtained
Thereby obtaining an optimal control voltage u n The expression is:
where P denotes an extraction matrix for extracting the disturbance phase from the predicted state vector and obtaining the control voltage by projection onto the TTM, expressed as:
P＝N -1 (1,0,10,...,...) (18)
thus, a control voltage u based on the LQG control model is obtained LQG ；
Step 11: then, a PI control model is established, a simplest integral controller is adopted, and a time domain expression of the PI control model is as follows:
u n ＝a*u n-1 +b*u e (19)
wherein u is e Is an error control signal, expressed as:
u e ＝u n -u n-1 (20)
obtaining PI control voltage u PI ；
Step 12: the PI control and the LQG control are connected in parallel to obtain a final control voltage, which is expressed as:
u o ＝u LQG +u PI (21)
step 13: final control voltage u o And the light beam is sent to the tilting mirror, and the tilting mirror is driven to generate corresponding mirror surface deflection, so that the light beam jitter signal is corrected.
2. The method for suppressing beam jitter according to claim 1, wherein the method comprises: the Control structure of the adaptive optical system is composed of a Wave Front Sensor (WFS), a Control Cell (CC), a Digital/Analog (D/A) converter and a time-to-stop Mirror (TTM).
3. The method for suppressing the optical beam jitter based on the LQG + PI hybrid control strategy according to claim 1, wherein: the beam-jitter control system is actually a closed-loop control implemented by WFS and TTM, the beam-jitter signal
Compensation correction via TTM
Obtaining a residual signal
The wave front detector detects the residual signal and sends the residual signal to the control unit, and the obtained control voltage u is applied to the TTM to generate surface shape change, thereby obtaining phase correction
And further realize the closed-loop control of the light beam dithering.
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