CN116248124A - Sigma-Delta modulator parameter design method based on LQG optimization algorithm - Google Patents

Abstract

This application introduces the LQG controller into the Sigma-Delta modulator system in order to improve the parameter mismatch robustness and disturbance suppression capability of the Sigma-Delta modulator. The invention adopts the LQG control method to better adjust the disturbance, noise, delay of the loop integrator and perturbation of each part parameter in the MEMS acceleration sensor system. The LQG control method is introduced into the Sigma-Delta modulator to effectively control the nonlinearity, uncertain factors, parameter perturbation and system delay contained in the modulator. LQG control mainly includes two parts, namely the estimation of the state variables and the calculation of the feedback gain matrix. Therefore, the present invention first uses Kalman filtering to estimate the state of the Sigma-Delta modulator system and perform filtering processing. Then, on the basis of the system state estimated by the Kalman filter, the linear quadratic optimal control theory is used to design the optimal control rate of the state feedback to optimally control the system.

Description

A Sigma-Delta Modulator Parameter Design Method Based on LQG Optimization Algorithm

Technical Field

The present application relates to the fields of analog-to-digital conversion, optimization control and signal processing, and in particular to a Sigma-Delta modulator parameter design method based on an LQG optimization algorithm.

Background Art

In the process of converting analog signals to digital signals, the performance indicators of ADC play an important role in the accuracy, resolution, noise shaping and other performances of the entire MEMS accelerometer. For example, the current communication system has extremely high requirements for the bandwidth of the analog-to-digital converter; the audio system also has high requirements for the accuracy of the analog-to-digital converter. Traditional analog-to-digital converters include dual-integral ADC, successive approximation ADC, pipeline ADC, etc. The conversion accuracy of these traditional analog-to-digital converters is generally not high and cannot meet the high-precision requirements of the system. Sigma-Delta ADC is a type of oversampling analog-to-digital converter. It is a high-precision and high-stability analog-to-digital converter, and has low requirements for the performance of the front-end anti-aliasing filter. In the research of high-precision MEMS accelerometers, compared with low-precision Nyquist rate ADCs, Sigma-Delta ADCs proportionally reduce the average noise power to achieve high accuracy and high linearity. In the MEMS accelerometer system, Sigma-Delta ADC includes micromechanical sensitive structure, front-end detection circuit and Sigma-Delta modulator, and the oversampling and noise shaping principles based on Sigma-Delta modulation technology are the main solutions for modern high-resolution device applications. In the capacitive MEMS accelerometer system, the micromechanical sensitive structure is modeled and analyzed using a second-order "spring-damper" mechanical system. Environmental factors (temperature, humidity, pressure, etc.) will cause the second-order model parameters of the sensitive unit to change, resulting in deviations between the actual model and the ideal model of the sensitive unit. This will prevent the movable mass block from being completely maintained in a balanced position, thereby reducing the performance of the entire MEMS accelerometer system. In addition, the entire closed-loop Sigma-Delta modulator system will inevitably be disturbed by external disturbances. If such disturbances can be eliminated in a timely manner, the performance of the capacitive MEMS accelerometer system can be further improved.

Summary of the invention

The present invention introduces an LQG controller into a Sigma-Delta modulator system in order to improve the parameter mismatch robustness and disturbance suppression capability of the Sigma-Delta modulator. LQG (Linear-Quadratic-Gaussian), i.e., linear quadratic Gaussian, is essentially a linear optimal control based on performance indicators.

The embodiment of the present application discloses a Sigma-Delta modulator parameter design method based on the LQG optimization algorithm, which comprises the following steps in sequence:

S1 uses Kalman filtering to estimate the state of the Sigma-Delta modulator system and perform filtering. The state equation is:

x _k =F _k+1 ·x _k +w _k (1)

In formula (1), _xk is the system state vector, which is obtained from the system's past behavior. _Fk+1,k is a known quantity, which represents the state transfer matrix of the state vector _xk from time k to time k+1. _wk represents Gaussian white noise with zero mean, and its corresponding variance matrix is _Qk .

Observation equation:

y _k =H _k ·x _k +v _k (2)

In formula (2), _yk represents the observation value at time k, _Hk represents the known observation matrix, _vk represents the zero-mean Gaussian white noise independent of _wk , and its corresponding variance is _Rk .

表示系统的先验状态估计，可由上一状态向量估计

得到，即：Based on the observed values y ₁ ,y ₂ ,…,y _k , find the least squares estimate of each component of the state vector x _k , assuming

Represents the prior state estimate of the system, which can be estimated from the previous state vector

Get, that is:

可通过式(4)计算得到：At this moment, the state estimate

It can be calculated by formula (4):

The state error vector is defined here as:

The steps of the Kalman filter algorithm can be summarized as follows:

Initialize the algorithm, that is, when k = 0, set the initial value of the state estimate to:

Initialization of posterior probability:

For k = 1, 2, ..., iterative calculation,

A priori state estimate:

Prior variance:

Kalman gain:

State estimate update:

Posterior variance update:

According to equations (1) and (3), the prior error value is:

S2 uses the state feedback method and combines the linear quadratic optimal control principle to solve the optimal state feedback gain of the system, realize the optimal control of the entire Sigma-Delta modulator system, and improve the dynamic performance of the system.

Assume there is a linear time-varying system:

In formula (7), x(t) represents the state quantity of the system, u(t) represents the control quantity of the system, y(t) represents the output quantity of the system, A, B, and C represent the known matrices of the system. In the optimal control theory, its linear quadratic performance index functional can be expressed as:

In formula (8), S is a semi-positive definite constant matrix, Q is a semi-positive definite weighted matrix, and R is a positive definite weighted symmetric matrix. The meaning of formula (8) is as follows: the first term represents the terminal state tracking error requirement of the entire system, which measures the requirement for the entire process control, and it is hoped that its value is as small as possible; the second term represents the requirement for control capability, which measures the minimum loss in the entire control process. The role of R is to ensure that the control quantity is within the actual control range and does not weaken the control capability. In actual engineering applications, the values of Q and R need to be set according to the requirements of system control performance. For the Sigma-Delta modulator system, its performance index does not include the system terminal term. For a given linear steady-state system, the performance index functional can be defined as:

In formula (9), _tf represents the terminal time, Q and R represent semi-positive definite symmetric matrix and positive definite symmetric matrix respectively, and u(t) represents the control rate. The control quantity u(t) with the minimum linear quadratic performance index in formula (9) can be expressed as:

u(t)＝-R ^-1 B ^T Px(t)＝-Kx(t) (10)

The state feedback gain of the optimal control system is:

K＝R ^-1 B ^T P (11)

P in formula (10) is the solution of the Riccati equation:

PA-A ^T P+PBR ^-1 B ^T PQ＝0 (12)

S3 combines the above Kalman filter and optimal controller design based on the separation theorem and the state feedback method, and finally obtains the LQG controller of the entire closed-loop Sigma-Delta modulator system.

The system state equation can be written as:

In the Sigma-Delta modulator system, D = 0, and since u(t) = -Kx(t), then

Formula (14) realizes the linear quadratic optimal control of the whole system based on state feedback. Through the input and output of the system, the Kalman filter is used to construct the state variables of the system. The Kalman filter not only has the function of state estimation, but also has the function of filtering, filtering out the noise caused by the external disturbance and parameter perturbation of the system.

LQG controller design, assuming that the linear time-varying system state equation is:

In formula (15), [A B] is controllable, [A C] is observable, w(t), V(t) represent Gaussian white noise with zero mean, w represents model noise, V represents observation noise, A, B, F, C represent the constant matrix of the system, and according to the minimum value theorem in linear quadratic optimal control, the optimal control rate of formula (15) can be derived as:

K _z = R ^-1 B ^T P (17)

为状态量x(t)的估计值，K_z为状态反馈中最优控制增益，P为式(12)方程的解，In formula (16) and (17)

is the estimated value of the state quantity x(t), K _z is the optimal control gain in the state feedback, P is the solution of equation (12),

For a specific Sigma-Delta modulator system, the role of the Kalman filter is to achieve noise filtering of the system and estimation and feedback of state variables. Assume that the state equation of a specific Sigma-Delta modulator system is as shown in equation (14). The state input and observed output of the system are used as the input of the Kalman filter. According to the Kalman filter theorem, the state estimation value of the system is:

In formula (18), L is the Kalman filter gain. According to the Kalman filter principle, we can get:

In formula (19), P ₀ is the solution of the following equation in formula (20):

系统状态估计值

和卡尔曼增益L，By using the Kalman filter program in the MATLAB toolbox to filter and estimate the system state, the system output estimate can be obtained.

System state estimate

and the Kalman gain L,

According to the Kalman filter theorem:

The Kalman filter principle diagram can be obtained through equation (21). Next, the optimal controller is designed based on the system state estimate. The premise of the linear quadratic optimal controller design is to obtain the optimal state feedback gain. Based on the given state estimate, the weighted matrices Q and R are the only two variables that can determine the system state feedback gain. Therefore, for the system in the form of equation (15), if the system represented by (15) is completely controllable, then under the action of the optimal control rate u(t) = -Kx(t), the state equation of the system is:

The selection of Q value needs to take into account the stability of the closed-loop system. Assume that the weighted matrix Q, R obtained by specific selection can calculate the solution P of the Riccati equation, thereby obtaining the optimal state feedback gain value K _z of the system.

The state estimation value of the system is obtained through the Kalman filter, and the optimal LQR controller is obtained according to the linear quadratic optimal control principle. The LQG control method is to combine the above two parts into an overall LQG controller. Finally, the design principle of the LQG controller is obtained.

Compared with the prior art, the main purpose of the present invention is to improve the performance of the Sigma-Delta modulator system, and to adopt the LQG control method to better regulate the disturbance, noise, delay of the loop integrator and the parameter perturbation of each part in the MEMS acceleration sensor system. The LQG control method is introduced into the Sigma-Delta modulator, and the nonlinearity, uncertainty factors and parameter perturbation and system delay contained in the modulator are effectively controlled. The LQG control mainly includes two parts, namely the estimation of state variables and the determination of the feedback gain matrix. Therefore, the present invention first adopts Kalman filtering to estimate the state of the Sigma-Delta modulator system and perform filtering processing. Then, based on the system state estimated by the Kalman filter, the linear quadratic optimal control theory is adopted to design the state feedback optimal control rate to optimally control the system. The simulation experiment results prove the effectiveness of the method proposed by the present invention.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

FIG1 is a block diagram of a system state feedback with a Kalman filter in an embodiment of the present invention;

FIG2 is a block diagram showing the principle of Kalman filtering in an embodiment of the present invention;

FIG3 is a block diagram showing the design principle of an LQG controller according to an embodiment of the present invention;

FIG4 is a schematic diagram of a Sigma-Delta modulator system based on an LQG controller according to an embodiment of the present invention;

FIG5 is a structural diagram of an LQG controller according to an embodiment of the present invention;

FIG6 is a noise power spectrum function density diagram with an LQG controller according to an embodiment of the present invention;

FIG. 7 shows a PSD diagram of the system with different spring coefficients according to an embodiment of the present invention.

FIG8 shows a PSD diagram of a fifth-order system with different spring coefficients in an embodiment of the present invention.

DETAILED DESCRIPTION

The following will describe the technical solutions in the embodiments of the present invention in detail in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

LQG control mainly includes two parts: Kalman filtering and linear quadratic optimal control. Therefore, the present invention first briefly describes these two parts of LQG.

1 Kalman filter theory

The Kalman filter is derived based on the system's previous state estimate and current state measurement to obtain the current state estimate, and then combines the error between the current state measurement and the current state estimate to re-derive a state quantity that is closest to the true value.

(1) Equation of state:

x _k =F _k+1 ·x _k +w _k (1)

In formula (1), _xk is the system state vector, which is obtained from the system's past behavior. _Fk+1,k is a known quantity, which represents the state transfer matrix of the state vector _xk from time k to time k+1. _wk represents Gaussian white noise with zero mean, and its corresponding variance matrix is _Qk .

(2) Observation equation:

y _k =H _k ·x _k +v _k (2)

In formula (2), _yk represents the observation value at time k, _Hk represents the known observation matrix, _vk represents the zero-mean Gaussian white noise independent of _wk , and its corresponding variance is _Rk .

表示系统的先验状态估计，可由上一状态向量估计

得到，即：Kalman filtering is the process of jointly solving the state equation and observation equation represented by equations (1) and (2) above through some optimization method. Specifically, it can be summarized as follows: based on the observation values y ₁ ,y ₂ ,…,y _k , the least squares estimate of each component of the state vector x _k is obtained. Assume

Represents the prior state estimate of the system, which can be estimated from the previous state vector

Get, that is:

可通过式(4)计算得到：At this moment, the state estimate

It can be calculated by formula (4):

The state error vector is defined here as:

The steps of the Kalman filter algorithm can be summarized as follows:

Initialize the algorithm, that is, when k = 0, set the initial value of the state estimate to:

Initialization of posterior probability:

For k = 1, 2, ..., iterative calculation

A priori state estimate:

Prior variance:

Kalman gain:

State estimate update:

Posterior variance update:

According to equations (1) and (3), the prior error value is:

From the introduction of the above Kalman filter algorithm, we can see that the Kalman algorithm can be easily implemented using a digital computer.

2 Linear quadratic optimal control

The optimal control method based on the minimum linear quadratic performance index is widely used. This optimal control method takes the integral of the system state quantity and control rate as the performance index function of the optimal control, and obtains the control variable corresponding to the minimum performance index function. This method is easy to construct optimal feedback control and easy to implement in actual engineering. Assume there is a linear time-varying system:

In formula (7), x(t) represents the state quantity of the system, u(t) represents the control quantity of the system, y(t) represents the output quantity of the system, and A, B, and C represent the known matrices of the system. In the optimal control theory, its linear quadratic performance index functional can be expressed as:

In formula (8), S is a semi-positive definite constant matrix, Q is a semi-positive definite weighted matrix, and R is a positive definite weighted symmetric matrix. The meaning of formula (8) is: the first term represents the terminal state tracking error requirement of the entire system, which measures the requirement for the control of the entire process, and it is hoped that its value is as small as possible; the second term represents the requirement for control capability, which measures the minimum loss in the entire control process. The role of R is to ensure that the control quantity is within the actual control range and does not weaken the control capability. In actual engineering applications, the values of Q and R need to be set according to the requirements of the system control performance. For the Sigma-Delta modulator system, its performance index does not include the system terminal term. For a given linear steady-state system, the performance index functional can be defined as:

In formula (9), _tf represents the terminal time, Q and R represent semi-positive definite symmetric matrix and positive definite symmetric matrix respectively, and u(t) represents the control rate. The control quantity u(t) with the minimum linear quadratic performance index in formula (9) can be expressed as:

u(t)＝-R ^-1 B ^T Px(t)＝-Kx(t) (10)

The state feedback gain of the optimal control system is:

K＝R ^-1 B ^T P (11)

P in formula (10) is the solution of the Riccati equation:

PA-A ^T P+PBR ^-1 B ^T PQ＝0 (12)

However, in the Sigma-Delta modulator system operation process, because the sensitive result is subject to external interference and factors such as self-parameter perturbation and internal disturbance, and in addition, its input signal is the signal difference of the input and output of the system, these factors will cause the whole closed-loop system to be impossible to realize more accurate control, and therefore, the dynamic performance of the whole system is not very high. In the present invention, in order to improve the dynamic performance of the whole system, the state feedback method is introduced into the Sigma-Delta modulator system. Specifically, the state estimation of the system is obtained on the basis of the Kalman filter, and various noise interferences are filtered out simultaneously, and the state feedback method is adopted to solve the system optimal state feedback gain in combination with the linear quadratic optimal control principle simultaneously, so as to realize the optimal control of the whole Sigma-Delta modulator system and improve the dynamic performance of the system.

3. Design of Sigma-Delta Modulator with LQG Controller

During the operation of the Sigma-Delta modulator system, due to the influence of various interference factors, it is inevitable that the stability, signal-to-noise ratio, resolution and accuracy of the system will be affected. In order to use the LQG control method to improve the performance of the entire Sigma-Delta modulator system, it is necessary to design an LQG controller in combination with the mathematical model of the MEMS accelerometer system. Specifically, the Kalman filter is first used to estimate the state of the Sigma-Delta modulator system. On the basis of the system state estimation value, the linear quadratic optimal control theory is used to design the optimal controller for the MEMS accelerometer system. Based on the separation theorem, the above-mentioned Kalman filter and optimal controller design are combined with the state feedback method, and finally the LQG controller of the entire closed-loop Sigma-Delta modulator system is obtained.

The system state equation given by the present invention can be written as:

In the Sigma-Delta modulator system of the present invention, D = 0, and since u(t) = -Kx(t), then

Formula (14) realizes the linear quadratic optimal control of the whole system on the basis of state feedback. However, the premise for the effectiveness of this method is that all state variables of the system must be measurable. Therefore, the present invention uses a Kalman filter to construct the state variables of the system through the input and output of the system. The Kalman filter in the present invention not only has a state estimation function, but also has a filtering function to filter out the noise caused by the external disturbance and parameter perturbation of the system. FIG1 shows a state feedback block diagram with a Kalman filter.

LQG controller design

Assume the state equation of the linear time-varying system is:

In formula (15), [A B] is controllable, [A C] is observable, w(t), V(t) represent Gaussian white noise with zero mean, w represents model noise, and V represents observation noise. A, B, F, C represent the constant matrix of the system. According to the minimum value theorem in linear quadratic optimal control, the optimal control rate of formula (15) can be derived as:

K _z = R ^-1 B ^T P (17)

为状态量x(t)的估计值，K_z为状态反馈中最优控制增益，P为式(12)方程的解。In formula (16) and (17)

is the estimated value of the state quantity x(t), _Kz is the optimal control gain in the state feedback, and P is the solution of equation (12).

For a specific Sigma-Delta modulator system, the role of the Kalman filter is to achieve noise filtering of the system and estimation and feedback of state variables. Assume that the state equation of a specific Sigma-Delta modulator system is as shown in equation (14), and the state input and observed output of the system are used as the input of the Kalman filter. According to the Kalman filter theorem, the state estimation value of the system is:

In formula (18), L is the Kalman filter gain. According to the Kalman filter principle, we can get:

In formula (19), P ₀ is the solution of the following equation in formula (20):

系统状态估计值

和卡尔曼增益L。为了接下来具体仿真验证LQG控制方法的有效性，这里给出简单计算范例。取系统的状态噪声方差矩阵为：By using the Kalman filter program in the MATLAB toolbox to filter and estimate the system state, the system output estimate can be easily obtained.

System state estimate

And Kalman gain L. In order to verify the effectiveness of the LQG control method in the following simulation, a simple calculation example is given here. The state noise variance matrix of the system is taken as:

QN = 3000. The variance matrix of the observation noise is:

The gain matrix of the Kalman filter can be obtained through MATLAB:

According to the Kalman filter theorem:

According to formula (21), the principle diagram of Kalman filtering can be obtained as shown in Figure 2.

Next, we design the optimal controller based on the system state estimate. The premise of the linear quadratic optimal controller design is to find the optimal state feedback gain. Based on the given state estimate, the weighted matrices Q and R are the only two variables that can determine the system state feedback gain. Therefore, for a system in the form of equation (15), if the system represented by (15) is completely controllable, then under the action of the optimal control rate u(t) = -Kx(t), the state equation of the system is:

The selection of Q value needs to take into account the stability of the closed-loop system. Assume that the weighted matrix Q, R obtained by specific selection can calculate the solution P of the Riccati equation, thereby obtaining the optimal state feedback gain value K _z of the system. The above-mentioned state estimation values of the system are obtained through the Kalman filter respectively, and the optimal (LQR) controller is obtained according to the linear quadratic optimal control principle. The LQG control method is to combine the above two parts into an overall LQG controller. Finally, the design principle block diagram of the LQG controller is given here, as shown in Figure 3.

In Figure 3, A, B, and C represent the constant matrices of the system, _Kz is the optimal state feedback gain of the system, and L is the Kalman gain.

4 Simulation Study of Sigma-Delta Modulator System Based on LQG Controller

In order to verify the effectiveness of the designed LQG controller, the present invention firstly provides a schematic diagram of the principle of a Sigma-Delta modulator system based on the LQG controller as shown in FIG4 .

In FIG4 , Gc(s) is an LQG controller, and its specific structure is shown in FIG5 .

According to Figure 5, the transfer function of the LQG controller can be obtained:

G _c (s)＝K(sI-A+BK+LC) ^-1 ·L (23)

In Figure 4, M(s) represents the mechanical sensitive structure of the MEMS acceleration sensor, and its expression is:

In formula (24), b is the damping coefficient of the second-order spring damping system, and k is the spring coefficient.

H(z) represents a third-order loop integrator, which is in the form of

Here are also the specific parameters required by the system during the simulation process, as shown in the following table:

parameter Numeric Mass of mass block [mg] 20 Damping coefficient b[Ns/m] 2.4×10 ^-3 Spring constant k[N/m] 100 AFE gain k ₀ [V/m] 4.55×10 ⁷ Comparator gain k ₁ 1 Feedback gain k ₂ [m/s ² ] 49.05 Input signal frequency [kHz] 128 Over frequency 64

According to the parameters in the above table, the fifth-order Sigma-Delta modulator system matrix A, B, C can be obtained, and the state equation of the fifth-order Sigma-Delta modulator can be obtained. According to formula (23), the LQG controller Gc(s) is derived. Substituting Gc(s) into the simulation model (4), the noise power spectral density function (PSD) diagram of the fifth-order Sigma-Delta modulator with the LQG controller can be finally obtained, as shown in Figure 6.

FIG6 also shows the PSD diagram of the pure fifth-order Sigma-Delta modulator. It can be seen from FIG6 that the system has a lower noise floor and a higher system signal-to-noise ratio due to the addition of the LQG controller.

In order to verify that the designed LQG controller has better robustness, the spring coefficients k are set to ks1 = 60 N/m, ks2 = 120 N/m, and ks3 = 240 N/m. Figures 7 and 8 show the PSD diagrams of the fifth-order Sigma-Delta modulator with an LQG controller and the PSD diagrams of the pure fifth-order Sigma-Delta modulator under different spring coefficients, respectively.

It can be seen from Figure 7 that the fifth-order Sigma-Delta modulator with LQG controller has better system robustness.

The following table shows the comparison of the LQG controller in improving the performance indicators of the fifth-order Sigma-Delta modulator.

It can be seen from the above table that the LQG controller designed in the present invention has better parameter mismatch robustness.

In summary, the present invention introduces the LQG controller into the fifth-order Sigma-Delta modulator to observe and compensate for the disturbance caused by the change of sensitive structural parameters. Based on the specific fifth-order Sigma-Delta modulator, the design ideas and design steps of the LQG controller are given. The designed LQG controller is used in the fifth-order Sigma-Delta modulator. Through simulation and result comparison, it is verified that the LQG controller can further improve the signal-to-noise ratio and suppress the noise floor of the fifth-order Sigma-Delta modulator. At the same time, it is verified that the LQG controller has good robustness in terms of Sigma-Delta modulator model parameter mismatch.

This implementation mode is only an exemplary description of this patent and does not limit its scope of protection. Personnel in this field may also make partial changes to it. As long as it does not exceed the spirit of this patent, it shall be regarded as an equivalent replacement of this patent and shall be within the scope of protection of this patent.

Claims (1)

1. A Sigma-Delta modulator parameter design method based on LQG optimization algorithm, characterized in that it comprises the following steps in sequence: S1 uses Kalman filtering to estimate the state of the Sigma-Delta modulator system and perform filtering. The state equation is: x _k =F _k+1 ·x _k +w _k (1) In formula (1), _xk is the system state vector, which is obtained from the system's past behavior. _Fk+1,k is a known quantity, which represents the state transfer matrix of the state vector _xk from time k to time k+1. _wk represents Gaussian white noise with zero mean, and its corresponding variance matrix is _Qk . Observation equation: y _k =H _k ·x _k +v _k (2) In formula (2), _yk represents the observation value at time k, _Hk represents the known observation matrix, _vk represents the zero-mean Gaussian white noise independent of _wk , and its corresponding variance is _Rk . Based on the observed values y ₁ ,y ₂ ,…,y _k , find the least squares estimate of each component of the state vector x _k , assuming

Represents the prior state estimate of the system, which can be estimated from the previous state vector

Get, that is:

At this moment, the state estimate

It can be calculated by formula (4):

The state error vector is defined here as:

The steps of the Kalman filter algorithm can be summarized as follows: Initialize the algorithm, that is, when k = 0, set the initial value of the state estimate to:

Initialization of posterior probability:

For k = 1, 2, ..., iterative calculation, A priori state estimate:

Prior variance:

Kalman gain:

State estimate update:

Posterior variance update:

According to equations (1) and (3), the prior error value is:

S2 uses the state feedback method and combines the linear quadratic optimal control principle to solve the optimal state feedback gain of the system, realize the optimal control of the entire Sigma-Delta modulator system, and improve the dynamic performance of the system. Assume there is a linear time-varying system:

In formula (7), x(t) represents the state quantity of the system, u(t) represents the control quantity of the system, y(t) represents the output quantity of the system, A, B, and C represent the known matrices of the system. In the optimal control theory, its linear quadratic performance index functional can be expressed as:

In formula (8), S is a semi-positive definite constant matrix, Q is a semi-positive definite weighted matrix, and R is a positive definite weighted symmetric matrix. The meaning of formula (8) is as follows: the first term represents the terminal state tracking error requirement of the entire system, which measures the requirement for the entire process control, and it is hoped that its value is as small as possible; the second term represents the requirement for control capability, which measures the minimum loss in the entire control process. The role of R is to ensure that the control quantity is within the actual control range and does not weaken the control capability. In actual engineering applications, the values of Q and R need to be set according to the requirements of system control performance. For the Sigma-Delta modulator system, its performance index does not include the system terminal term. For a given linear steady-state system, the performance index functional can be defined as:

In formula (9), _tf represents the terminal time, Q and R represent semi-positive definite symmetric matrix and positive definite symmetric matrix respectively, and u(t) represents the control rate. The control quantity u(t) with the minimum linear quadratic performance index in formula (9) can be expressed as: u(t)＝-R ^-1 B ^T Px(t)＝-Kx(t) (10) The state feedback gain of the optimal control system is: K＝R ^-1 B ^T P (11) P in formula (10) is the solution of the Riccati equation: PA-A ^T P+PBR ^-1 B ^T PQ＝0 (12) S3 combines the above Kalman filter and optimal controller design based on the separation theorem and the state feedback method, and finally obtains the LQG controller of the entire closed-loop Sigma-Delta modulator system. The system state equation can be written as:

In the Sigma-Delta modulator system, D = 0, and since u(t) = -Kx(t), then

Formula (14) realizes the linear quadratic optimal control of the whole system based on state feedback. Through the input and output of the system, the Kalman filter is used to construct the state variables of the system. The Kalman filter not only has the function of state estimation, but also has the function of filtering, filtering out the noise caused by the external disturbance and parameter perturbation of the system. LQG controller design, assuming that the linear time-varying system state equation is:

In formula (15), [A B] is controllable, [A C] is observable, w(t), V(t) represent Gaussian white noise with zero mean, w represents model noise, V represents observation noise, A, B, F, C represent the constant matrix of the system, and according to the minimum value theorem in linear quadratic optimal control, the optimal control rate of formula (15) can be derived as:

K _z = R ^-1 B ^T P (17) In formula (16) and (17)

is the estimated value of the state quantity x(t), K _z is the optimal control gain in the state feedback, P is the solution of equation (12), For a specific Sigma-Delta modulator system, the role of the Kalman filter is to achieve noise filtering of the system and estimation and feedback of state variables. Assume that the state equation of a specific Sigma-Delta modulator system is as shown in equation (14). The state input and observed output of the system are used as the input of the Kalman filter. According to the Kalman filter theorem, the state estimation value of the system is:

In formula (18), L is the Kalman filter gain. According to the Kalman filter principle, we can get:

In formula (19), P ₀ is the solution of the following equation in formula (20):

By using the Kalman filter program in the MATLAB toolbox to filter and estimate the system state, the system output estimate can be obtained.

System state estimate

and the Kalman gain L, According to the Kalman filter theorem:

The Kalman filter principle diagram can be obtained through equation (21). Next, the optimal controller is designed based on the system state estimate. The premise of the linear quadratic optimal controller design is to obtain the optimal state feedback gain. Based on the given state estimate, the weighted matrices Q and R are the only two variables that can determine the system state feedback gain. Therefore, for the system in the form of equation (15), if the system represented by (15) is completely controllable, then under the action of the optimal control rate u(t) = -Kx(t), the state equation of the system is:

The selection of Q value needs to take into account the stability of the closed-loop system. Assume that the weighted matrix Q, R obtained by specific selection can calculate the solution P of the Riccati equation, thereby obtaining the optimal state feedback gain value K _z of the system. The state estimation value of the system is obtained through the Kalman filter, and the optimal LQR controller is obtained according to the linear quadratic optimal control principle. The LQG control method is to combine the above two parts into an overall LQG controller. Finally, the design principle of the LQG controller is obtained.

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