CN105067893B - Solution resistance flexible measurement method based on conductance cell second order model - Google Patents

Abstract

The invention relates to a solution resistance soft measurement method based on a second-order system model of a conductivity cell, which belongs to the technical field of solution conductivity soft measurement. It is characterized in that the measurement of conductivity is converted into the parameter estimation of the second-order system model of the conductance cell considering the influence of lead distributed capacitance and electric double layer capacitance, specifically the solution resistance, lead distributed capacitance and electric double layer capacitance to be estimated in a small period For a constant approximation, a parameter state space model is established; a sinusoidal combination signal is used to excite the conductance cell system, and based on the sampling signal obtained by the high-speed A/D for the excitation signal and the system response signal, the Kalman model constructed according to the parameter state space model is started. After the filter is recursively calculated for N steps in each hour period, the estimated values of solution resistance, lead distributed capacitance and electric double layer capacitance in each hour period are obtained. The effects and benefits of the invention are strong anti-interference ability, high estimation precision, complete real-time numerical calculation, and are suitable for industrial online application of conductivity measurement.

Description

Soft-sensing method for solution resistance based on second-order system model of conductivity cell

technical field

The invention belongs to the technical field of solution conductivity soft measurement, and relates to a method for estimating the parameters of the second-order equivalent resistance-capacitance system of a conductivity cell. And the response data, reconstruct the soft sensor method of the solution resistance estimate value through dynamic filtering.

Background technique

Solution conductivity is an important electrochemical parameter. Electrode conductivity measurement is a commonly used measurement method for solution conductivity, which is mainly affected by polarization effects, capacitance effects and temperature. The influence of temperature can be eliminated by constant temperature method or compensation method, and the polarization effect can be eliminated by AC or pulse excitation, so the capacitive effect becomes the key factor affecting the measurement of solution conductivity. With the development of soft-sensing technology, when the soft-sensing method is applied to the measurement of solution conductivity, the main idea is to build the capacitive effect as interference into the equivalent resistance-capacitance system model of the conductivity cell, and easily analyze the excitation signal and system response signal. The measured variable is measured, and the estimated value of the solution resistance (conductivity) is obtained by the method of parameter estimation. The capacitance effect is more complicated, mainly including lead distributed capacitance and electric double layer capacitance. If only the distributed capacitance of the leads is considered and the influence of the electric double layer capacitance is ignored, then the first-order equivalent resistance-capacitance system model of the conductance cell is established. This model is relatively simple, but it misses the objective existence of the electric double layer capacitance. the elements of. The literature "Cui Pengfei, Zhang Liyong, Zhong Chongquan, Li Dan. Numerical simulation of multi-frequency square wave excitation RC decoupling soft measurement. Journal of Instrumentation, 2010, 31(1): 154-160" using multiple frequency AC The square wave excites the conductance cell respectively. Based on the first-order equivalent resistance-capacitance system model of the conductance cell, the relationship between the excitation signal, the response DC voltage signal, the solution resistance and the lead distributed capacitance is established, and the non-linear least square method is used to Estimating the two parameters of solution resistance and lead distributed capacitance can weaken the influence of various uncertainties in the measurement, but the optimization solution adopts the steepest descent method, which requires iterative calculation, and there is a problem of uncertain number of iterations. If the soft-sensing method with square wave excitation is extended to the second-order equivalent resistance-capacitance system model of the conductivity cell considering the influence of the lead distributed capacitance and the electric double layer capacitance, the complexity will make the method difficult to implement. The patent document "Zhou Kaidi, Zhang Liyong, Ling Jingwei, Zhong Chongquan, Li Dan. Resistor-capacitance decoupling soft measurement method based on amplitude-phase characteristic detection (ZL 201010173466.3)" aims at the first-order equivalent of conductance cell considering the influence of lead distributed capacitance The resistance-capacitance system model adopts sinusoidal signal excitation, uses the multi-point sampling value of the response signal to fit its function form, and then obtains the system amplitude and phase characteristic parameters, and then obtains The estimated value of the two parameters of solution resistance and lead distributed capacitance is obtained; in this method, the acquisition of system amplitude and phase characteristic parameters still adopts the nonlinear least square method, and the optimization solution adopts the steepest descent method, which also has the problem of uncertain iteration times. The patent document "Zhang Liyong, Yang Chunhua, Zhou Kaidi, Li Xiong, Wang Jiayue, Li Zugang. Conductivity second-order resistance-capacitance coupling network parameter estimation method (ZL 201210383286.7)" is a soft sensing method based on amplitude and phase characteristics detection by a conductivity cell The first-order equivalent resistance-capacitance system model is extended to the second-order equivalent resistance-capacitance system model, that is, the influence of the lead distributed capacitance and the electric double layer capacitance is considered at the same time; when the method is implemented, two sinusoidal signals with different frequencies need to be used to respectively excite the conductance After obtaining the amplitude and phase characteristic parameters of the system at two different frequencies respectively, it is necessary to use the subspace confidence region method to optimize and solve the estimated values of solution resistance, lead distributed capacitance and electric double layer capacitance, and the complexity of the solution is greatly increased. The patent document "Zhang Liyong, Zhong Chongquan, Lu Wei, Yang Chunhua, Wang Jiayue, Li Xiong. A dynamic filter estimation method for the parameters of the first-order resistance-capacitance system of conductivity (ZL201310002557.4)" is aimed at a conductivity cell that only considers the influence of lead distributed capacitance. The first-order equivalent resistance-capacitance system model uses a sinusoidal signal to excite the conductivity cell, and reconstructs the estimated value of the solution resistance and lead distributed capacitance in real time and accurately through the method of dynamic system dynamic filtering; this method has strong anti-interference ability, The steps of the recursive operation are completely determined, and the real-time estimation of the parameters can be obtained with high precision, but the influence of the electric double layer capacitance is ignored.

Contents of the invention

The technical problem to be solved by the present invention is: for the second-order equivalent resistance-capacitance system model of the conductance cell considering the influence of the distributed capacitance of the lead wire and the electric double layer capacitance, how to use the AC signal to excite the conductance cell, how to dynamically filter the conductance cell through the dynamic system, real-time , to accurately reconstruct the estimated value of the solution resistance.

Technical scheme of the present invention is:

The measurement problem of conductivity is attributed to the estimation problem of the parameters of the second-order equivalent resistance-capacitance system model of the conductivity cell. The transfer function model of the second-order equivalent resistance-capacitance system of the conductance cell is established as

Among them: R _x is the solution resistance, C _x is the electric double layer capacitance, C _p is the distributed capacitance of the lead, R ₁ is the voltage divider resistance, U(s) is the Laplace transform of the system excitation signal u(t), Y (s) is the Laplace transform of the system response signal y(t), and

m ₁ =R _x C _x (2)

m ₂ =R ₁ R _x C _x C _p (3)

m ₃ =R ₁ C _p +R ₁ C _x +R _x C _x (4)

The sampling period is denoted as T _s , and the formula (1) is differentiated by using the backward difference method, so that The z transfer function of the second-order equivalent resistance-capacitance system of the conductance cell can be obtained as

Formula (5) can be written as

Let u(k) and y(k) denote the sampling values of the system excitation signal u(t) and the system response signal y(t) at time k respectively, and transform the formula (6) into the time domain difference equation model as

in:

The estimated parameters R _x , C _x and C _p change with time during the whole measurement process, but they can be approximated to be constant in each hour segment. Let the duration of each hour segment be NT _s , and N is a positive integer. From formula (2) to formula (4), it can be seen that m ₁ , m ₂ and m ₃ are also constant in each small period; from formula (8) to formula (11), it can be seen that a ₁ , a ₂ and b ₀ , b ₁ is also constant in each hour period. Within one hour, the state equation (12) is established with a ₁ , a ₂ , b ₀ , and b ₁ as the state variables, and the equation (7) is rewritten into the observation equation (13), and a new parameter state space expression can be obtained:

θ(k+1)=θ(k) (12)

y(k)=hT(k)θ(k)+ ⁿ (k) (13)

in:

h ^T (k)=[y(k-1) -y(k-2) u(k) -u(k-1)] (14)

θ(k)=[a ₁ a ₂ b ₀ b ₁ ] ^T (15)

n(k) is the observation noise, and its variance is

For the system shown in formula (12) and formula (13), the Kalman filter formula (16) - formula (20)

S(k)=P(k-1) (18)

P(k)=[IK(k) ^hT (k)]P(k-1) (19)

Recursive N steps, you can get the estimated value of θ in this hour period The estimated values of m ₁ , m ₂ and m ₃ can be obtained from formula (8) - formula (11) with

Then, the estimated value of solution resistance R _x , electric double layer capacitance C _x and lead distributed capacitance C _p can be obtained from formula (2) - formula (4) with

Depend on The conductivity of the solution can be obtained.

After obtaining the estimated values of solution resistance, lead distributed capacitance and electric double layer capacitance for the first hour period, next, repeat the Kalman filter formula (16)-equation (20) N steps in each subsequent hour period , to obtain the estimated values of a ₁ , a ₂ , b ₀ , b ₁ in the respective small periods, and then solve C _p , C _x and R _x in the respective small periods by formula (24)-formula (26) estimated value.

During the measurement, the alternating sinusoidal signal is selected as the excitation signal, which can effectively suppress the polarization effect inside the conductivity cell. In addition, when selecting the excitation signal, it is also required that the excitation signal can fully excite all modes of the system. The conductance cell system is a high-order system in nature. If only a single frequency sinusoidal signal is used to excite the conductance cell, it is obvious that the mode of the system cannot be fully excited, which will lead to a decrease in the accuracy of parameter estimation. Therefore, the sinusoidal combination signal is selected as the motivating signal. The sinusoidal composite signal requires that the initial phases of each sinusoidal signal included are the same, which may be set to 0rad, and the angular frequencies are w, 2w,...,2 ^q w, q is an integer and q≥1, that is

The sinusoidal combination signal can effectively suppress the polarization effect of the conductivity cell, and can well excite each mode of the system, so that the parameter estimation result of the dynamic filter is closer to the true value.

The effect and benefits of the present invention are that the soft measurement method based on dynamic filtering has strong anti-interference ability, the number of steps of recursive calculation is completely determined, and real-time estimation of parameters can be obtained with high precision, which is suitable for the industrial measurement of conductivity Apply online.

Description of drawings

The accompanying drawing is a measurement block diagram of the solution resistance soft measurement method based on the second-order system model of the conductivity cell.

In the figure: R ₁ is the voltage divider resistance, R _x is the solution resistance, C _p is the lead distributed capacitance, C _x is the electric double layer capacitance, u(t) is the sinusoidal combination excitation signal, y(t) is the system response signal, u(k) is the sampling value of high-speed A/D to u(t), and y(k) is the sampling value of high-speed A/D to y(t).

detailed description

The specific embodiments of the present invention will be described in detail below in conjunction with the technical solutions and accompanying drawings.

According to the specific time-varying conditions of the solution resistance, lead distributed capacitance and electric double layer capacitance parameters in practical applications, the length of the small segment is set. Usually, it should not be too large on the basis of ensuring the convergence of the filter.

A sinusoidal combined excitation signal u(t) is used to excite the conductance cell system, and the system response signal is y(t). Sampling u(t) and y(t) via high-speed A/D according to the sampling period T _s respectively to obtain sampling signals u(k) and y(k), k=1, 2, . . .

While sampling, start the Kalman filter formula (16) - formula (20), after N steps of recursive calculation, the lead distributed capacitance C in the first hour period can be obtained by formula (24) - formula (26) _p , the estimated values of electric double layer capacitance _Cx and solution resistance _Rx .

Next, repeat the Kalman filter formula (16) - formula (20) N steps in each subsequent hour period, and then solve the C _p , Estimated values of C _x and R _x .

Claims (1)

1. A solution resistance soft measurement method based on conductance cell second-order system model, it is characterized in that following steps: (1) conductance cell is equivalent to the second-order resistance-capacitance system that considers the influence of lead wire distributed capacitance and electric double layer capacitance, From its time-domain difference equation model formula (7): y(k)=a ₁ y(k-1)-a ₂ y(k-2)+b ₀ u(k)-b ₁ u(k-1), (7) in: m ₁ =R _x C _x , m ₂ =R ₁ R _x C _x C _p , m ₃ =R ₁ C _p +R ₁ C _x +R _x C _x , R _x is the solution resistance, C _x is the electric double layer Capacitance, C _p is the distributed capacitance of the lead, R ₁ is the voltage divider resistor, T _s is the sampling period; Under the approximation that the parameters R _x , C _x and C _p to be estimated in the small period NT _s are constant, N is a positive integer; a parameter state space model is established with a ₁ , a ₂ , b ₀ , and b ₁ as state variables ( 12) and formula (13): θ(k+1)=θ(k), (12) y(k)=hT(k)θ(k)+ ⁿ (k), (13) Where: h ^T (k)=[y(k-1) -y(k-2) u(k) -u(k-1)], θ(k)=[a ₁ a ₂ b ₀ b ₁ ] ^T , n(k) is the observation noise; (2) Using a sinusoidal combination excitation signal formed by superimposing sinusoidal signals with angular frequencies w, 2w, ..., 2 ^q w <mrow><mi>u</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><munderover><mo>&amp;Sigma;</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>q</mi></munderover><mi>s</mi><mi>i</mi><mi>n</mi><mrow><mo>(</mo><msup><mn>2</mn><mi>i</mi></msup><mi>w</mi><mi>t</mi><mo>)</mo></mrow><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>27</mn><mo>)</mo></mrow></mrow> Excited conductance cell system, where: q is an integer and q≥1; according to the sampling period T _s , the sinusoidal combination excitation signal and the system response signal are sampled by high-speed A/D respectively, and the sampled signals u(k) and y(k ), k=1, 2, ...; (3) While sampling, start the Kalman filter constructed according to the parameter state space model formula (12) and formula (13), based on the sampling signals u(k) and y(k) in the first hour period, After N steps of recursive operation, the estimated value of θ in the first hour period is obtained Depend on with The estimated values of lead distributed capacitance C _p , electric double layer capacitance C _x and solution resistance R _x in the first hour period are obtained by formula (24), formula (25) and formula (26) with <mrow><msub><mover><mi>C</mi><mo>^</mo></mover><mi>p</mi></msub><mo>=</mo><mfrac><msub><mover><mi>m</mi><mo>^</mo></mover><mn>2</mn></msub><mrow><msub><mover><mi>m</mi><mo>^</mo></mover><mn>1</mn></msub><msub><mi>R</mi><mn>1</mn></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>T</mi><mi>s</mi></msub><msub><mover><mi>a</mi><mo>^</mo></mover><mn>2</mn></msub></mrow><mrow><msub><mi>R</mi><mn>1</mn></msub><msub><mover><mi>b</mi><mo>^</mo></mover><mn>1</mn></msub></mrow></mfrac><mo>,</mo><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>24</mn><mo>)</mo></mrow></mrow> <mrow><msub><mover><mi>C</mi><mo>^</mo></mover><mi>x</mi></msub><mo>=</mo><mfrac><mn>1</mn><msub><mi>R</mi><mn>1</mn></msub></mfrac><mrow><mo>(</mo><msub><mover><mi>m</mi><mo>^</mo></mover><mn>3</mn></msub><mo>-</mo><msub><mover><mi>m</mi><mo>^</mo></mover><mn>1</mn></msub><mo>-</mo><msub><mi>R</mi><mn>1</mn></msub><msub><mover><mi>C</mi><mo>^</mo></mover><mi>p</mi></msub><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mo>&amp;lsqb;</mo><mrow><mo>(</mo><msub><mover><mi>a</mi><mo>^</mo></mover><mn>1</mn></msub><mo>-</mo><msub><mover><mi>b</mi><mo>^</mo></mover><mn>1</mn></msub><mo>)</mo></mrow><msub><mover><mi>b</mi><mo>^</mo></mover><mn>1</mn></msub><mo>-</mo><mrow><mo>(</mo><msub><mover><mi>b</mi><mo>^</mo></mover><mn>0</mn></msub><mo>+</mo><msub><mover><mi>b</mi><mo>^</mo></mover><mn>1</mn></msub><mo>)</mo></mrow><msub><mover><mi>a</mi><mo>^</mo></mover><mn>2</mn></msub><mo>&amp;rsqb;</mo><msub><mi>T</mi><mi>s</mi></msub></mrow><mrow><msub><mover><mi>b</mi><mo>^</mo></mover><mn>1</mn></msub><mrow><mo>(</mo><msub><mover><mi>b</mi><mo>^</mo></mover><mn>0</mn></msub><mo>-</mo><msub><mover><mi>b</mi><mo>^</mo></mover><mn>1</mn></msub><mo>)</mo></mrow><msub><mi>R</mi><mn>1</mn></msub></mrow></mfrac><mo>,</mo><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>25</mn><mo>)</mo></mrow></mrow> <mrow><msub><mover><mi>R</mi><mo>^</mo></mover><mi>x</mi></msub><mo>=</mo><mfrac><msub><mover><mi>m</mi><mo>^</mo></mover><mn>1</mn></msub><msub><mover><mi>C</mi><mo>^</mo></mover><mi>x</mi></msub></mfrac><mo>=</mo><mfrac><mrow><msubsup><mover><mi>b</mi><mo>^</mo></mover><mn>1</mn><mn>2</mn></msubsup><msub><mi>R</mi><mn>1</mn></msub></mrow><mrow><mo>(</mo><msub><mover><mi>a</mi><mo>^</mo></mover><mn>1</mn></msub><mo>-</mo><msub><mover><mi>b</mi><mo>^</mo></mover><mn>1</mn></msub><mo>)</mo><msub><mover><mi>b</mi><mo>^</mo></mover><mn>1</mn></msub><mo>-</mo><mo>(</mo><msub><mover><mi>b</mi><mo>^</mo></mover><mn>0</mn></msub><mo>+</mo><msub><mover><mi>b</mi><mo>^</mo></mover><mn>1</mn></msub><mo>)</mo><msub><mover><mi>a</mi><mo>^</mo></mover><mn>2</mn></msub></mrow></mfrac><mo>;</mo><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>26</mn><mo>)</mo></mrow></mrow> in: with are estimated values of m ₁ , m ₂ and m ₃ , respectively; (4) Next, in each subsequent hour period, based on the sampled signals u(k) and y(k) in the respective hour period, the Kalman filter is repeatedly run for N steps, and after obtaining the a ₁ , After the estimated values of a ₂ , b ₀ and b ₁ are obtained, the estimated values of C _p , C _x and R _x in the respective small segments can be obtained from formula (24), formula (25) and formula (26).