kalman

Kalman filter design, Kalman estimator

Syntax

[kest,L,P] = kalman(sys,Qn,Rn,Nn) [kest,L,P] = kalman(sys,Qn,Rn,Nn,sensors,known) [kest,L,P,M,Z] = kalman(sys,Qn,Rn,...,type)

Description

kalman designs a Kalman filter or Kalman state estimator given a state-space model of the plant and the process and measurement noise covariance data. The Kalman estimator provides the optimal solution to the following continuous or discrete estimation problems.

Continuous-Time Estimation

Given the continuous plant

$\begin{array}{l} \dot{x} = A x + B u + G w (state equation) \\ y = C x + D u + H w + v (measurement equation) \end{array}$

with known inputs u, white process noise w, and white measurement noise v satisfying

$E (w) = E (v) = 0, E (w w^{T}) = Q, E (v v^{T}) = R, E (w v^{T}) = N$

construct a state estimate $\hat{x} (t)$ that minimizes the steady-state error covariance
$P = \lim_{t \to \infty}$ $E ({x - \hat{x}} {x - \hat{x}}^{T})$

The optimal solution is the Kalman filter with equations

$\begin{array}{l} \dot{\hat{x}} = A \hat{x} + B u + L (y - C \hat{x} - D u) \\ [\begin{matrix} \hat{y} \\ \hat{x} \end{matrix}] = [\begin{matrix} C \\ I \end{matrix}] \hat{x} + [\begin{matrix} D \\ 0 \end{matrix}] u \end{array}$

The filter gain L is determined by solving an algebraic Riccati equation to be

$L = (P C^{T} + \bar{N}) {\bar{R}}^{- 1}$

where

$\begin{array}{l} \bar{R} = R + H N + N^{T} H^{T} + H Q H^{T} \\ \bar{N} = G (Q H^{T} + N) \end{array}$

and P solves the corresponding algebraic Riccati equation.

The estimator uses the known inputs u and the measurements y to generate the output and state estimates $\hat{y}$ and $\hat{x}$ . Note that $\hat{y}$ estimates the true plant output

$y = C x + D u + H w + v$

Discrete-Time Estimation

Given the discrete plant

$\begin{array}{l} x [n + 1] = A x [n] + B u [n] + G w [n] \\ y [n] = C x [n] + D u [n] + H w [n] + v [n] \end{array}$

and the noise covariance data

$E (w [n] w {[n]}^{T}) = Q, E (v [n] v {[n]}^{T}) = R, E (w [n] v {[n]}^{T}) = N$

The estimator has the following state equation:

$\hat{x} [n + 1 | n] = A \hat{x} [n | n - 1] + B u [n] + L (y [n] - C \hat{x} [n | n - 1] - D u [n])$

The gain matrix L is derived by solving a discrete Riccati equation to be

$L = (A P C^{T} + \bar{N}) {(C P C^{T} + \bar{R})}^{- 1}$

where

$\begin{array}{l} \bar{R} = R + H N + N^{T} H^{T} + H Q H^{T} \\ \bar{N} = G (Q H^{T} + N) \end{array}$

There are two variants of discrete-time Kalman estimators:

The current estimator generates output estimates $\hat{y} [n | n]$ and state estimates $\hat{x} [n | n]$ using all available measurements up to $y [n]$ . This estimator has the output equation

$[\begin{matrix} \hat{y} [n | n] \\ \hat{x} [n | n] \end{matrix}] = [\begin{matrix} (I - M_{y}) C \\ I - M_{x} C \end{matrix}] \hat{x} [n | n - 1] + [\begin{matrix} (I - M_{y}) D & M_{y} \\ - M_{x} D & M_{x} \end{matrix}] [\begin{matrix} u [n] \\ y [n] \end{matrix}],$
where the innovation gains M_x and M_y are defined as:

$\begin{matrix} M_{x} = P C^{T} {(C P C^{T} + \bar{R})}^{- 1}, \\ M_{y} = (C P C^{T} + H Q H^{T} + H N) {(C P C^{T} + \bar{R})}^{- 1} . \end{matrix}$
M_x updates the prediction $\hat{x} [n | n - 1]$ using the new measurement $y [n]$ .

$\hat{x} [n | n] = \hat{x} [n | n - 1] + M_{x} \underset{i n n o v a t i o n}{\underset{︸}{(y [n] - C \hat{x} [n | n - 1] - D u [n])}} .$
When H = 0, $M_{y} = C M_{x}$ and $y [n | n] = C x [n | n] + D u [n]$ .
The delayed estimator generates output estimates $\hat{y} [n | n - 1]$ and state estimates $\hat{x} [n | n - 1]$ using measurements only up to y_v[n–1]. This estimator is easier to implement inside control loops and has the output equation

$[\begin{matrix} \hat{y} [n | n - 1] \\ \hat{x} [n | n - 1] \end{matrix}] = [\begin{matrix} C \\ I \end{matrix}] \hat{x} [n | n - 1] + [\begin{matrix} D & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} u [n] \\ y [n] \end{matrix}]$

[kest,L,P] = kalman(sys,Qn,Rn,Nn) creates a state-space model kest of the Kalman estimator given the plant model sys and the noise covariance data Qn, Rn, Nn (matrices Q, R, N described in Description). sys must be a state-space model with matrices $A, [B G], C, [D H]$ .

The resulting estimator kest has inputs $[u; y]$ and outputs $[\hat{y}; \hat{x}]$ (or their discrete-time counterparts). You can omit the last input argument Nn when N = 0.

The function kalman handles both continuous and discrete problems and produces a continuous estimator when sys is continuous and a discrete estimator otherwise. In continuous time, kalman also returns the Kalman gain L and the steady-state error covariance matrix P. P solves the associated Riccati equation.

[kest,L,P] = kalman(sys,Qn,Rn,Nn,sensors,known) handles the more general situation when

Not all outputs of sys are measured.
The disturbance inputs w are not the last inputs of sys.

The index vectors sensors and known specify which outputs y of sys are measured and which inputs u are known (deterministic). All other inputs of sys are assumed stochastic.

[kest,L,P,M,Z] = kalman(sys,Qn,Rn,...,type) specifies the estimator type for discrete-time plants sys. The type argument is either 'current' (default) or 'delayed'. For discrete-time plants, kalman returns the estimator and innovation gains L and M and the steady-state error covariances

$\begin{array}{l} P = \lim_{n \to \infty} E (e [n | n - 1] e [^{n | n - 1] T}), e [n | n - 1] = x [n] - x [n | n - 1] \\ Z = \lim_{n \to \infty} E (e [n | n] e [^{n | n] T}), e [n | n] = x [n] - x [n | n] \end{array}$

Examples

See LQG Design for the x-Axis and Kalman Filtering for examples that use the kalman function.

Limitations

The plant and noise data must satisfy:

(C,A) detectable
$\bar{R} > 0$ and $\bar{Q} - \bar{N} {\bar{R}}^{- 1} {\bar{N}}^{T} \geq 0$
$(A - \bar{N} {\bar{R}}^{- 1} C, \bar{Q} - \bar{N} {\bar{R}}^{- 1} {\bar{N}}^{T})$ has no uncontrollable mode on the imaginary axis (or unit circle in discrete time) with the notation

$\begin{array}{l} \bar{Q} = G Q G^{T} \\ \bar{R} = R + H N + N^{T} H^{T} + H Q H^{T} \\ \bar{N} = G (Q H^{T} + N) \end{array}$

References

[1] Franklin, G.F., J.D. Powell, and M.L. Workman, Digital Control of Dynamic Systems, Second Edition, Addison-Wesley, 1990.

[2] Lewis, F., Optimal Estimation, John Wiley & Sons, Inc, 1986.

[3] Deshpande, A.S., "Bridging a Gap in Applied Kalman Filtering: Estimating Outputs When Measurements Are Correlated with the Process Noise." IEEE Control Systems Magazine, Vol. 37, Number 3, 2017, pp. 87–93.

Documentation

kalman

Syntax

Description

Examples

Limitations

References

See Also

Topics

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