代做ECE5550: Applied Kalman Filtering SIMULTANEOUS STATE AND PARAMETER ESTIMATION USING KALMAN FILTERS

ECE5550: Applied Kalman Filtering

SIMULTANEOUS STATE AND PARAMETER ESTIMATION USING KALMAN FILTERS

9.1: Parameters versus states

■ Until now, we have assumed that the state-space model of the system whose state we are estimating is known and constant.

■ However, the system model may not be entirely known: We may wish to adapt numeric values within the model to better match the model’s behavior. to the true system’s behavior.

■ Also, certain values within the system may change very slowly over  the lifetime of the system—it would be good to track those changes.

■ For example, consider a battery cell. Its state-of-charge can traverse its entire range within minutes. However, its internal resistance might change as little as 20% in a decade or more of regular use.

• The quantities that tend to change quickly comprise the state of the system, and

• The quantities that tend to change slowly comprise the

time-varying parameters of the system.

■ We know that Kalman filters may be used to estimate the state of a dynamic system given known parameters and noisy measurements.

■ We may also use (nonlinear) Kalman filters to estimate parameters given a known state and noisy measurements.

■ In this section of notes we first consider how to estimate the parameters of a system if its state is known.

■ Next, we consider how to simultaneously estimate both the state and parameters of the system using two different approaches.

The generic approach to parameter estimation

■ We denote the true parameters of a particular model by θ .

■ We will use Kalman filtering techniques to estimate the parameters

much like we have estimated the state. Therefore, we require a model of the dynamics of the parameters.

■ By assumption, parameters change very slowly, so we model them as constant with some small perturbation:

θk = θk−1 + rk−1 .

■ The small white noise input rk is fictitious, but models the slow drift in  the parameters of the system plus the infidelity of the model structure.

■ The output equation required for Kalman-filter system identification must be a measurable function of the system parameters. We use

dk = hk (xk, uk,θ , ek ),

where h(·) is the output equation of the system model being     identified, and ek models the sensor noise and modeling error.

■ Note that dk is usually the same measurement as zk , but we maintain

a distinction here in case separate outputs are used. Then,

Dk = {d0, d1 , . . . , dk }. Also, note that ek and vk often play the same role, but are considered distinct here.

■ We also slightly revise the mathematical model of system dynamics

xk = fk−1(xk−1 , uk−1,θ,wk−1)

z k = hk (xk, uk,θ,vk ),

to explicitly include the parameters θ in the model.

■ Non-time-varying numeric values required by the model may be embedded within f (·) and h(·), and are not included in θ .

9.2: EKF for parameter estimation

■ Here, we show how to use EKF for parameter estimation.

■ As always, we proceed by deriving the six essential steps of sequential inference.

EKF step 1a: Parameter estimate time update.

■ The parameter prediction step is approximated as

■ This makes sense, since the parameters are assumed constant. EKF step 1b: Error covariance time update.

■ The covariance prediction step is accomplished by first computing θ˜ k −.k— .

■ We then directly compute the desired covariance

■ The time-updated covariance has additional uncertainty due to the fictitious noise “driving” the parameter values.

EKF step 1c: Output estimate.

■ The system output is estimated to be

d(^)k = E[h(xk, uk,θ , ek ) | Dk — 1]

≈ hk (xk , uk , θ(^)k— , e-k ).

■ That is, it is assumed that propagatingθ(^)k— and the mean estimation

error is the best approximation to estimating the output.

EKF step 2a: Estimator gain matrix.

■ The output prediction error may then be approximated

using again a Taylor-series expansion on the first term.

■ From this, we can compute such necessary quantities as

■ These terms may be combined to get the Kalman gain

■ Note, by the chain rule of total differentials,

■ But,

■ The derivative calculations are recursive in nature, and evolve over time as the state evolves.

■ The term dx0/dθ is initialized to zero unless side information gives a better estimate of its value.

■ To calculateC(^)k(θ) for any specific model structure, we require methods

to calculate all of the above the partial derivatives for that model.

EKF step 2b: State estimate measurement update.

■ The fifth step is to compute the a posteriori state estimate by

updating the a priori estimate using the estimator gain and the output prediction error dk − d(^)k

EKF step 2c: Error covariance measurement update.

■ Finally, the updated covariance is computed as

■ EKF for parameter estimation is summarized in a later table.

Notes:

■ We initialize the parameter estimate with our best information re. the parameter value: θ(^)0(十) = E[θ0].

■ We initialize the parameter estimation error covariance matrix:

■ We also initialize dx0 /dθ = 0 unless side information is available.