All-vanadium redox flow battery SOC estimation method fusing forgetting factor least square method and unscented Kalman filtering

Abstract

The invention discloses an all-vanadium redox flow battery SOC estimation method fusing a forgetting factor least square method and unscented Kalman filtering, and relates to the technical field of electrochemical energy storage state monitoring, and the method comprises the steps: collecting voltage and current data of an all-vanadium redox flow battery, and carrying out the data processing; establishing a Thevenin equivalent circuit model of the all-vanadium redox flow battery; performing parameter identification on the equivalent circuit model by adopting a forgetting factor recursive least square method, and optimizing a forgetting factor and an initial parameter value by adopting a starfish optimization algorithm, so that an error between a terminal voltage value simulated by the equivalent circuit model and an actual voltage value is minimum; based on the parameters of the real-time identification equivalent circuit model, the unscented Kalman filter algorithm is combined to carry out joint estimation on the polarization voltage and the SOC of the battery, so that the estimated terminal voltage error is minimum, and the estimated value of the SOC of the all-vanadium redox flow battery is further obtained. Therefore, by adopting the method, accurate estimation of the SOC of the battery is realized, so that the electric quantity condition of the battery is judged, and guidance and reference are provided for field operation and maintenance personnel.

Description

Technical Field

[0001] The invention relates to the technical field of electrochemical energy storage state monitoring, and in particular to an all-vanadium liquid flow battery SOC estimation method integrating a forgetting factor least squares method and an unscented Kalman filter. Background Art

[0002] With the continuous advancement of the "dual carbon" goals, building a new power system has become an important direction for the development of my country's power industry at this stage. As the core technology and key equipment supporting the new power system, energy storage technology is experiencing rapid development. Among them, all-vanadium liquid flow batteries have shown great potential in the field of large-scale long-term energy storage with their excellent peak-shaving and valley-filling capabilities and the advantages of maintaining grid stability.

[0003] Accurate estimation of the state of charge (SOC) of the all-vanadium liquid flow battery is the key to determining the remaining power of the battery and can provide a reasonable charge and discharge control strategy for the battery system. Accurate SOC estimation can not only effectively prevent over-charge and over-discharge of the battery and extend the battery life, but also reduce system maintenance costs and further improve the economy and operating efficiency of the battery.

[0004] In recent years, significant progress has been made in the field of electrochemical energy storage state monitoring technology. The current battery SOC estimation methods are mainly divided into three categories: traditional methods, modern filtering algorithms, and other methods. The application of new sensing technologies, data processing methods, and intelligent algorithms has enabled more accurate and efficient battery state estimation methods to be realized. Based on this, this paper provides a vanadium liquid flow battery SOC estimation method that integrates the forgetting factor least squares method and the unscented Kalman filter. Summary of the invention

[0005] The purpose of the present invention is to provide a method for estimating the SOC of an all-vanadium liquid flow battery by integrating the least squares method with the forgetting factor and the unscented Kalman filter, which can accurately estimate the SOC of the battery, thereby judging the power status of the battery and providing guidance and reference for on-site operation and maintenance personnel.

[0006] To achieve the above object, the present invention provides a method for estimating SOC of an all-vanadium liquid flow battery by integrating a forgetting factor least squares method and an unscented Kalman filter, comprising the following steps:

[0007] S1, collect voltage and current data of all-vanadium liquid flow battery and process the data;

[0008] S2. Establish the Thevenin equivalent circuit model of all-vanadium liquid flow battery;

[0009] S3. Use the forgetting factor recursive least squares method to identify the parameters of the equivalent circuit model, and use the starfish optimization algorithm to optimize the forgetting factor and the initial parameter values, so that the error between the terminal voltage value simulated by the equivalent circuit model and the actual voltage value is minimized;

[0010] S4. Based on the real-time identification of the parameters of the equivalent circuit model, and combined with the unscented Kalman filter algorithm, jointly estimate the battery polarization voltage and SOC, so that the error of the estimated terminal voltage is minimized, and then obtain the estimated value of the SOC of the all-vanadium redox flow battery.

[0011] Preferably, in step S1, it includes obtaining the voltage and current changes of the battery through the hybrid pulse power characteristic test method, collecting the charge and discharge data of the all-vanadium redox flow battery under low current conditions, and performing data processing; using the low current test method to obtain the relationship between the OCV and SOC of the all-vanadium redox flow battery, and constructing a functional relationship through polynomial fitting.

[0012] Preferably, in step S2, it includes using a voltage source to represent the OCV associated with SOC and temperature, using the ohmic resistance to represent the influence of the internal current excitation of the battery, and using a parallel RC network to simulate the transient dynamics involved in the battery, and reflecting the internal mechanism of the battery by establishing a Thevenin equivalent circuit model.

[0013] Preferably, in step S3, the forgetting factor recursive least squares method includes using the error value between the simulated terminal voltage and the actual terminal voltage as the objective function, and using the starfish optimization algorithm to optimize the forgetting factor and the initial parameters, as follows:

[0014] First, initialize the population size, the maximum number of iterations, and the starfish position vector of the starfish, and map the forgetting factor and the parameters to the starfish position vector;

[0015] Secondly, use the identified model terminal voltage error as the fitness function, and calculate the fitness value corresponding to each starfish position;

[0016] Then, according to the random number and the global probability value, judge whether to enter the exploration stage or the exploitation stage, and then update the position vector and check the boundary;

[0017] Then, iteratively update the position vector, calculate the new fitness value, and retain the current optimal forgetting factor and parameters until the maximum number of iterations condition is met.

[0018] Preferably, in the exploration stage, calculate the rotation angle and energy. When the dimension is greater than 5, select some dimensions to update the position. When the dimension is less than or equal to 5, randomly update the position of a single dimension;

[0019] During the exploitation stage, calculate the distance between the starfish and the global optimal position, and update the position vector according to the distance and a random number; after updating the position vector, check the boundary and execute the regeneration mechanism to prevent falling into the local optimum.

[0020] Preferably, in step S3, it includes estimating the parameters of the all-vanadium redox flow battery regression model in real time based on voltage and current data, and identifying the parameters of the equivalent circuit model by introducing a forgetting factor, as follows:

[0021]

[0022] In the formula, G k is the gain vector, is the input vector, P k , P k-1 are the covariance matrices at time k and time k-1 respectively, are the parameter estimates at time k and time k-1 respectively, λ is the forgetting factor, y k is the actual output voltage of the system.

[0023] Preferably, after identifying the parameters of the equivalent circuit model, based on the hybrid pulse power characteristic test method and the forgetting factor recursive least squares method, identify the open-circuit voltage, ohmic internal resistance, polarization internal resistance, and polarization capacitance parameters at each SOC state through voltage and current data, and establish the relationship between each parameter and SOC to simulate the dynamic characteristics of the all-vanadium redox flow battery at each state of charge.

[0024] Preferably, step S4 includes establishing a nonlinear state space model based on the Thevenin equivalent circuit model of the all-vanadium redox flow battery, as follows:

[0025]

[0026] V t,k = g(x k , u k ) + ν k = V oc,k + V p,k + R s u k + ν k ;

[0027] In the formula, x = [V p , SOC] T is the state variable, the input vector u is the battery current I L ; ω k represents the process noise obeying the normal distribution with variance Q, ν k is the measurement noise obeying the normal distribution with variance R, f(·) and g(·) are nonlinear functions, u kThe current value at time k, V p,k , V t,k are the polarization voltage and the terminal voltage at time k, respectively, V oc,k is the open-circuit voltage at time k, R s is the ohmic resistance.

[0028] Preferably, the polarization voltage and the SOC of the battery are jointly estimated by combining the unscented Kalman filter algorithm, including the following steps:

[0029] First, initialize the mean value and covariance matrix of the state variables;

[0030] Secondly, obtain 2n + 1 Sigma sampling points through unscented transformation, predict the state quantity and covariance matrix, calculate the mean value of the system prediction, and calculate the cross-covariance matrix and covariance matrix at the same time;

[0031] Then, calculate the Kalman gain matrix and update the state matrix and error covariance matrix.

[0032] Preferably, the state matrix and error covariance matrix are updated as follows:

[0033]

[0034] In the formula, K k is the Kalman gain matrix, is the predicted state estimate value, Y k is the actual value of the observed quantity, is the mean value of the predicted value of the observed quantity, is the covariance matrix of the observed quantity, P k / (k-1) is the predicted value of the error covariance matrix, P k are the updated state matrix and error covariance matrix, respectively.

[0035] Therefore, the SOC estimation method of the all-vanadium redox flow battery adopting the above-mentioned method of fusing the forgetting factor least squares method and the unscented Kalman filter has the following technical effects: by fusing the intelligent optimization algorithm and the equivalent circuit model, the calculation cost is effectively reduced, the equivalent circuit model parameters are expressed by a polynomial fitting function, the simulation accuracy of the physical model is greatly improved, and combined with the advanced filtering technology, the accurate estimation of the state of charge of the battery is realized.

[0036] Next, through the drawings and embodiments, the technical solutions of the present invention will be further described in detail. Description of the Drawings

[0037] Figure 1 is a flowchart of an SOC estimation method for an all-vanadium redox flow battery that fuses the forgetting factor least squares method and the unscented Kalman filter;

[0038] Figure 2 Schematic diagram of the Thevenin first-order RC equivalent circuit model of a vanadium redox flow battery in an embodiment of a method for estimating the state of charge (SOC) of a vanadium redox flow battery that combines the forgetting factor least squares method and the unscented Kalman filter;

[0039] Figure 3 Flowchart of optimizing the parameters of the FFRLS by SFOA in an embodiment of a method for estimating the SOC of a vanadium redox flow battery that combines the forgetting factor least squares method and the unscented Kalman filter. Detailed implementation manners

[0040] The present invention can be more specifically explained by the following embodiments. The purpose of disclosing the present invention is to protect all changes and improvements within the scope of the present invention. The present invention is not limited to the following embodiments.

[0041] As Figure 1 shown, the present invention provides a method for estimating the SOC of a vanadium redox flow battery that combines the forgetting factor least squares method and the unscented Kalman filter. The specific steps include:

[0042] S1. Collect the voltage and current data of the vanadium redox flow battery and perform data processing.

[0043] S2. Establish the Thevenin equivalent circuit model of the vanadium redox flow battery.

[0044] S3. Use the forgetting factor recursive least squares method (FFRLS) to identify the parameters of the equivalent circuit model, and use the starfish optimization algorithm (SFOA) to optimize the forgetting factor and initial parameter values of the FFRLS, so that the error between the terminal voltage value simulated by the equivalent circuit model and the actual voltage value is minimized.

[0045] Among them, the hybrid pulse power characteristic test method is a method widely used for parameter identification of the battery equivalent circuit model. Usually, a series of pulse currents are applied at different SOCs, and the dynamic characteristics of the battery are analyzed through the dynamic current, so as to establish an equivalent circuit model to simulate the battery changes. As Figure 2 shown, the Thevenin first-order RC equivalent circuit model of the vanadium redox flow battery can be represented by establishing an equation through Kirchhoff's law as follows:

[0046] V t = V OC + V p + I L R s ;

[0047]

[0048] In the formula, I L is the battery current, which is positive during charging and negative during discharging; Vp , V t respectively represent the polarization voltage and the terminal voltage of the all-vanadium redox flow battery; V OC is the open-circuit voltage; R s is the ohmic internal resistance; C p is the polarization capacitance; R p is the polarization resistance. The polarization voltage can be approximately discretized by an exponential function to obtain the discrete-time form as:

[0049]

[0050] where Δt is the time interval for identifying the model parameters, and t is the current time.

[0051] For practical applications, the equivalent circuit model equation is discretized using the Laplace transform and represented by a transfer function as:

[0052]

[0053] where s is the Laplace operator.

[0054] Then, the bilinear transform is used to map the above transfer function equation based on the s-plane to the z-plane to obtain the z-plane transfer function equation:

[0055]

[0056]

[0057] where c 1 , c 2 , c 3 are coefficients related to the calculation of the model parameters, and z -1 is the delay operator.

[0058] Considering that the OCV of the battery changes relatively slowly, within a sufficiently small time difference, the change in V oc can be neglected. Therefore, through transformation, the voltage of the all-vanadium redox flow battery can be expressed as follows:

[0059]

[0060] where θ k are respectively the known input vector and the parameter vector to be estimated that describe the state of the system at time k.

[0061] To estimate the parameters θ k of the all-vanadium redox flow equivalent circuit model in real time, the FFRLS algorithm is used for calculation as follows:

[0062]

[0063] In the formula, G k is the gain vector, is the input vector, P k is the covariance matrix, is the parameter estimation value, λ is the forgetting factor, y k is the actual output value of the system, and the voltage is taken as the output here.

[0064] The open-circuit voltage, ohmic internal resistance, polarization internal resistance, and polarization capacitance parameters under each SOC state are identified through voltage and current data, and the relationships between these parameters and SOC are established, so as to simulate the dynamic characteristics of the all-vanadium redox flow battery under each state of charge. Then, after obtaining the parameter θ k through the solution, the open-circuit voltage, ohmic internal resistance, polarization capacitance, and polarization internal resistance parameters in the first-order RC equivalent circuit model are calculated respectively through the corresponding mathematical relationship expressions as follows:

[0065] θ k = [(1 - c 1 )V oc,k c 1 c 2 c 3 T = [θ 1 θ 2 θ 3 θ 4 T ;

[0066]

[0067] To minimize the model error, the error value between the simulated terminal voltage and the actual terminal voltage is taken as the objective function here, and the SFOA algorithm is used to optimize the forgetting factor and the initial parameter values of the FFRLS algorithm, as Figure 3 shown, and the specific steps include:

[0068] (1) Initialize the population size N and the maximum number of iterations T of the starfish in SFOA max and the starfish position vector, where the forgetting factor λ and the parameter θ in the FFRLS model are mapped to the starfish position vector.

[0069] (2) Take the FFRLS identification model terminal voltage error (mean square error MSE) as the fitness function, and calculate the fitness value corresponding to each starfish position.

[0070] (3) According to the random number and the global probability value G p , judge whether to enter the exploration stage or the exploitation stage.

[0071] (4) In the exploration stage, calculate the rotation angle ω and the energy E t ​​, when the dimension D > 5, select some dimensions to update the position, and when the dimension D ≤ 5, randomly update the position of a single dimension; in the exploitation stage, calculate the distance between the starfish and the global optimal position, and update the position vector according to the distance and a random number; after updating the position vector, check the boundary and execute the regeneration mechanism to prevent falling into the local optimum;

[0072] (5) Iteratively update the position vector, calculate the new fitness value, and retain the current optimal FFRLS forgetting factor λ and parameter θ until the maximum iteration number condition T is satisfied max . At this time, output the optimal FFRLS forgetting factor λ * and parameter θ * , perform equivalent circuit model parameter identification, so as to obtain the optimal identification result.

[0073] S4. Based on the FFRLS algorithm, identify the parameters such as capacitance and resistance in the equivalent circuit model in real time, and combine the unscented Kalman filter (UKF) algorithm to jointly estimate the battery polarization voltage and SOC, so that the error of the estimated terminal voltage is minimized, thereby realizing the accurate estimation of the SOC of the all-vanadium redox flow battery.

[0074] Among them, the SOC representing the internal state of the battery has obvious nonlinearity and time-variation, and its definition is as follows:

[0075]

[0076] In the formula, SOC k represents the SOC of the battery at time k, C n represents the rated capacity, Δt represents the sampling time interval, and η represents the Coulomb efficiency, which is regarded as a constant 1 here.

[0077] The nonlinear state space model of the all-vanadium redox flow battery can be expressed as follows:

[0078]

[0079] V t,k = g(x k , u k ) + ν k = V oc,k + V p,k + R s u k + ν k ;

[0080] In the formula, x = [V p , SOC] T is the state variable, the input vector u is the load current I L , ω k represents the process noise obeying the normal distribution with variance Q, ν kDenote the measurement noise that follows a normal distribution with variance R. f(·) and g(·) are non - linear functions, and u k is the current value at time k.

[0081] Combined with the state - space model of the all - vanadium redox flow battery, the UKF algorithm is used for state estimation. The specific implementation steps are as follows:

[0082] (1) Initialize the mean value of the state variable x and the covariance matrix P, and the expressions are as follows:

[0083]

[0084]

[0085] In the formula, E is the expected value, and the subscript 0 represents the initial state.

[0086] (2) Obtain 2n + 1 Sigma sampling points through unscented transformation, as follows:

[0087]

[0088] In the formula, X i,k-1 is the constructed point set, λ 1 is the scale correction factor, and the subscript i is the sampling point.

[0089] (3) Predict the state quantity and the covariance matrix, as follows:

[0090]

[0091] In the formula, f(X k-1,i , u k-1 ) is the system state equation; P k / (k-1) are the predicted values of the system state quantity and the covariance matrix respectively, and they are obtained by weighted summation of the predicted values X k / (k-1),i at the Sigma points; the covariance matrix P k / (k-1) also includes the process noise covariance Q k , which is used to describe the randomness and uncertainty of the system.

[0092] (4) Calculate the mean value of the system prediction, as follows:

[0093]

[0094] In the formula, g(X k-1,i , u k-1 ) is the observation equation.

[0095] (5) Calculate the cross - covariance matrix and the covariance matrix, as follows:

[0096]

[0097] In the formula, is the covariance matrix of the observed quantity; is the cross-covariance matrix between the state quantity and the observed quantity; ω c,i is the covariance weight, which is the weighting coefficient for each Sigma point in the unscented Kalman filter; Y k / (k-1),i is the predicted value of the observed quantity after the i-th Sigma point is mapped through the observation equation; is the average value of the predicted values of the observed quantity.

[0098] (6) Calculate the Kalman gain matrix K as follows:

[0099]

[0100] (7) Update the state matrix and the error covariance matrix as follows:

[0101]

[0102] In the formula, is the updated state matrix.

[0103] Based on the established battery model and data samples, the estimation accuracy of the model can be effectively verified, thereby providing support for improving the accuracy of the SOC estimation of the all-vanadium redox flow battery.

[0104] Therefore, the SOC estimation method of the all-vanadium redox flow battery adopting the above method combining the forgetting factor least squares method and the unscented Kalman filter can effectively reduce the calculation cost, greatly improve the simulation accuracy of the physical model, and achieve accurate estimation of the battery state of charge.

[0105] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions of the present invention or make equivalent replacements, and these modifications or equivalent replacements do not make the modified technical solutions deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for estimating SOC of an all-vanadium liquid flow battery by integrating the least squares method with the forgetting factor and the unscented Kalman filter, characterized in that: The following steps are involved: S1, collect voltage and current data of all-vanadium liquid flow battery and process the data; S2. Establish the Thevenin equivalent circuit model of all-vanadium liquid flow battery; S3, using the forgetting factor recursive least squares method to identify the parameters of the equivalent circuit model, and using the starfish optimization algorithm to optimize the forgetting factor and the initial parameter value, so that the error between the terminal voltage value simulated by the equivalent circuit model and the actual voltage value is minimized; S4. Based on the real-time identification of the parameters of the equivalent circuit model, the battery polarization voltage and SOC are jointly estimated in combination with the unscented Kalman filter algorithm, so that the estimated terminal voltage error is minimized, and then the SOC of the all-vanadium liquid flow battery is estimated.

2. The method for estimating SOC of an all-vanadium liquid flow battery integrating the forgetting factor least squares method and the unscented Kalman filter according to claim 1 is characterized in that: Step S1 includes obtaining the voltage and current changes of the battery through a mixed pulse power characteristic test method, collecting the charge and discharge data of the all-vanadium liquid flow battery under low current conditions, and performing data processing; The low current test method was used to obtain the relationship between OCV and SOC of the all-vanadium liquid flow battery, and the functional relationship was constructed through polynomial fitting.

3. The method for estimating SOC of an all-vanadium liquid flow battery integrating the forgetting factor least squares method and the unscented Kalman filter according to claim 1 is characterized in that: In step S2, a voltage source is used to represent the OCV associated with the SOC and temperature, an ohmic resistor is used to represent the influence of the internal current excitation of the battery, a parallel RC network is used to simulate the transient dynamics involved in the battery, and a Thevenin equivalent circuit model is established to reflect the internal mechanism of the battery.

4. The method for estimating SOC of an all-vanadium liquid flow battery integrating the forgetting factor least square method and the unscented Kalman filter according to claim 1, characterized in that: In step S3, the forgetting factor recursive least square method includes taking the error value between the simulated terminal voltage and the actual terminal voltage as the objective function and optimizing the forgetting factor and initial parameters using the starfish optimization algorithm as follows: First, initialize the starfish population size, maximum number of iterations, and starfish position vector, and map the forgetting factor and parameters to the starfish position vector; Secondly, the identified model terminal voltage error is used as the fitness function to calculate the fitness value corresponding to each starfish position; Next, based on the random number and the global probability value, determine whether to enter the exploration phase or the exploitation phase, and then update the position vector and check the boundary; Then, the position vector is iteratively updated, a new fitness value is calculated, and the current optimal forgetting factor and parameters are retained until the maximum number of iterations is met.

5. The method for estimating SOC of an all-vanadium liquid flow battery integrating the least square method with the forgetting factor and the unscented Kalman filter according to claim 4, characterized in that: include: In the exploration phase, the rotation angle and energy are calculated. When the dimension is greater than 5, some dimensions are selected to update the position. When the dimension is less than or equal to 5, the position of a single dimension is randomly updated. In the exploitation phase, the distance between the starfish and the global optimal position is calculated, and the position vector is updated according to the distance and the random number; after the position vector is updated, the boundary is checked and a regeneration mechanism is executed to prevent falling into the local optimum.

6. The method for estimating SOC of an all-vanadium liquid flow battery integrating the least square method with the forgetting factor and the unscented Kalman filter according to claim 1, characterized in that: In step S3, the parameters of the regression model of the all-vanadium liquid flow battery are estimated in real time based on the voltage and current data, and the parameters of the equivalent circuit model are identified by introducing a forgetting factor, as follows: In the formula, G k is the gain vector, is the input vector, P k , P k-1 are the covariance matrices at time k and time k-1 respectively, are the parameter estimates at time k and time k-1 respectively, λ is the forgetting factor, y k is the actual output voltage of the system.

7. The method for estimating SOC of an all-vanadium liquid flow battery integrating the forgetting factor least square method and the unscented Kalman filter according to claim 6, characterized in that: After identifying the parameters of the equivalent circuit model, based on the hybrid pulse power characteristic test method and the forgetting factor recursive least squares method, the open circuit voltage, ohmic internal resistance, polarization internal resistance and polarization capacitance parameters under each SOC state are obtained through voltage and current data identification, and the relationship between each parameter and SOC is established to simulate the dynamic characteristics of the all-vanadium liquid flow battery under various charge states.

8. The method for estimating SOC of an all-vanadium liquid flow battery integrating the forgetting factor least square method and the unscented Kalman filter according to claim 1, characterized in that: Step S4 includes establishing a nonlinear state space model based on the Thevenin equivalent circuit model of the all-vanadium liquid flow battery, as follows: V t,k =g(x k ,u k )+ν k =V oc,k +V p,k +R s u k +ν k ; In the formula, x = [V p ,SOC] T is the state variable, and the input vector u is the battery current I L ;ω k represents the process noise that follows a normal distribution with variance Q, ν k represents the measurement noise that follows a normal distribution with variance R, f(·) and g(·) are nonlinear functions, and u k is the current value at time k, V p,k 、V t,k are the polarization voltage and terminal voltage at time k, V oc,k is the open circuit voltage at time k, R s is the ohm resistance.

9. The method for estimating SOC of an all-vanadium liquid flow battery integrating the forgetting factor least square method and the unscented Kalman filter according to claim 8, characterized in that: The battery polarization voltage and SOC are jointly estimated by combining the unscented Kalman filter algorithm, including the following steps: First, initialize the mean and covariance matrices of the state variables; Secondly, 2n+1 Sigma sampling points are obtained through lossless transformation, the state quantity and covariance matrix are predicted, and the average value of the system prediction is calculated, and the cross-covariance matrix and covariance matrix are calculated at the same time; Then, the Kalman gain matrix is ​​calculated, and the state matrix and error covariance matrix are updated.

10. The method for estimating SOC of an all-vanadium liquid flow battery integrating the forgetting factor least square method and the unscented Kalman filter according to claim 9, characterized in that: Update the state matrix and error covariance matrix as follows: In the formula, K k is the Kalman gain matrix, is the estimated value of the predicted state, Y k is the actual value of the observed quantity, is the average value of the predicted value of the observation, is the covariance matrix of the observed quantity, P k / (k-1) is the predicted value of the error covariance matrix, P k are the updated state matrix and error covariance matrix respectively.

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