Form state estimator given estimator gain

`est = estim(sys,L) `

est = estim(sys,L,sensors,known)

`est = estim(sys,L) `

produces
a state/output estimator `est`

given the plant state-space
model `sys`

and the estimator gain `L`

.
All inputs *w* of `sys`

are assumed
stochastic (process and/or measurement noise), and all outputs *y* are
measured. The estimator `est`

is returned in state-space
form (SS object).

For a continuous-time plant `sys`

with equations

$$\begin{array}{l}\dot{x}=Ax+Bw\\ y=Cx+Dw\end{array}$$

`estim`

uses the following equations to generate
a plant output estimate $$\widehat{y}$$ and a state estimate $$\widehat{x}$$, which are estimates
of *y*(*t*)=*C* and *x*(*t*),
respectively:

$$\begin{array}{l}\dot{\widehat{x}}=A\widehat{x}+L(y-C\widehat{x})\\ \left[\begin{array}{c}\widehat{y}\\ \widehat{x}\end{array}\right]=\left[\begin{array}{c}C\\ I\end{array}\right]\widehat{x}\end{array}$$

For a discrete-time plant `sys`

with the following
equations:

$$\begin{array}{l}x[n+1]=Ax[n]+Bw[n]\\ y[n]=Cx[n]+Dw[n]\end{array}$$

`estim`

uses estimator equations similar to
those for continuous-time to generate a plant output estimate $$y[n|n-1]$$ and a state estimate $$x[n|n-1]$$, which are estimates
of *y*[*n*] and *x*[*n*],
respectively. These estimates are based on past measurements up to *y*[*n-1*].

`est = estim(sys,L,sensors,known) `

handles more general plants `sys`

with both known
(deterministic) inputs *u* and stochastic inputs *w*,
and both measured outputs *y* and nonmeasured outputs *z*.

$$\begin{array}{l}\dot{x}=Ax+{B}_{1}w+{B}_{2}u\\ \left[\begin{array}{c}z\\ y\end{array}\right]=\left[\begin{array}{c}{C}_{1}\\ {C}_{2}\end{array}\right]x+\left[\begin{array}{c}{D}_{11}\\ {D}_{21}\end{array}\right]w+\left[\begin{array}{c}{D}_{12}\\ {D}_{22}\end{array}\right]u\end{array}$$

The index vectors `sensors`

and `known`

specify
which outputs of `sys`

are measured (*y*),
and which inputs of `sys`

are known (*u*).
The resulting estimator `est`

, found using the following
equations, uses both *u* and *y* to
produce the output and state estimates.

$$\begin{array}{l}\dot{\widehat{x}}=A\widehat{x}+{B}_{2}u+L(y-{C}_{2}\widehat{x}-{D}_{22}u)\\ \left[\begin{array}{c}\widehat{y}\\ \widehat{x}\end{array}\right]=\left[\begin{array}{c}{C}_{2}\\ I\end{array}\right]\widehat{x}+\left[\begin{array}{c}{D}_{22}\\ 0\end{array}\right]u\end{array}$$

Consider a state-space model `sys`

with seven
outputs and four inputs. Suppose you designed a Kalman gain matrix *L* using
outputs 4, 7, and 1 of the plant as sensor measurements and inputs
1, 4, and 3 of the plant as known (deterministic) inputs. You can
then form the Kalman estimator by

sensors = [4,7,1]; known = [1,4,3]; est = estim(sys,L,sensors,known)

See the function `kalman`

for
direct Kalman estimator design.

You can use the functions `place`

(pole placement)
or `kalman`

(Kalman filtering) to design an adequate
estimator gain *L*. Note that the estimator poles
(eigenvalues of *A-LC*) should be faster than the
plant dynamics (eigenvalues of *A*) to ensure accurate
estimation.