Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

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.

Was this topic helpful?