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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.