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
with known inputs u, white process noise w, and white measurement noise v satisfying
construct a state estimate
that minimizes the steady-state error covariance
The optimal solution is the Kalman filter with equations
The filter gain L is determined by solving an algebraic Riccati equation to be
where
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
. Note that
estimates the true plant output
Discrete-Time Estimation
Given the discrete plant
and the noise covariance data
The estimator has the following state equation:
The gain matrix L is derived by solving a discrete Riccati equation to be
and
where
There are two variants of discrete-time Kalman estimators:

The current estimator generates output estimates
measurements up to
and state estimates
using all available
. This estimator has the output equation
where the innovation gain M is defined as
M updates the prediction

using the new measurement
The delayed estimator generates output estimates
.
and state estimates
using measurements
only up to yv[n-1]. This estimator is easier to implement inside control loops and has the output equation
[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
The resulting estimator kest has inputs
and outputs
.
(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 or
sys are assumed stochastic.
[kest,L,P,M,Z] = kalman(sys,Qn,Rn,...,type) specifies the estimator type for discrete-time
plants sys. The string type 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
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

and

has no uncontrollable mode on the imaginary axis (or unit circle in discrete time) with
the notation
References
[1] Franklin, G.F., J.D. Powell, and M.L. Workman, Digital Control of Dynamic Systems, Second Edition, AddisonWesley, 1990.
[2] Lewis, F., Optimal Estimation, John Wiley & Sons, Inc, 1986.