Form Linear-Quadratic-Gaussian (LQG) servo controller

`C = lqgtrack(kest,k)`

C = lqgtrack(kest,k,'2dof')

C = lqgtrack(kest,k,'1dof')

C = lqgtrack(kest,k,...CONTROLS)

`lqgtrack`

forms a Linear-Quadratic-Gaussian
(LQG) servo controller with integral action for the loop shown in
the following figure. This compensator ensures that the output *y* tracks
the reference command *r* and rejects process disturbances *w* and
measurement noise *v*. `lqgtrack`

assumes
that *r* and *y* have the same length.

Always use positive feedback to connect the LQG servo controller *C* to
the plant output *y*.

`C = lqgtrack(kest,k)`

forms a two-degree-of-freedom
LQG servo controller `C`

by connecting the Kalman
estimator `kest`

and the state-feedback gain `k`

,
as shown in the following figure. `C`

has inputs $$[r;y]$$ and generates
the command $$u=-K\left[\widehat{x};{x}_{i}\right]$$, where$$\widehat{x}$$ is the Kalman
estimate of the plant state, and *x _{i}* is
the integrator output.

The size of the gain matrix `k`

determines
the length of *x _{i}*.

The two-degree-of-freedom LQG servo controller state-space equations are

$$\begin{array}{l}\left[\begin{array}{c}\dot{\widehat{x}}\\ {\dot{x}}_{i}\end{array}\right]=\left[\begin{array}{cc}A-B{K}_{x}-LC+LD{K}_{x}& -B{K}_{i}+LD{K}_{i}\\ 0& 0\end{array}\right]\left[\begin{array}{c}\widehat{x}\\ {x}_{i}\end{array}\right]+\left[\begin{array}{cc}0& L\\ I& -I\end{array}\right]\left[\begin{array}{c}r\\ y\end{array}\right]\\ u=\left[\begin{array}{cc}-{K}_{x}& -{K}_{i}\end{array}\right]\left[\begin{array}{c}\widehat{x}\\ {x}_{i}\end{array}\right]\end{array}$$

The syntax `C = lqgtrack(kest,k,'2dof')`

is
equivalent to `C = lqgtrack(kest,k)`

.

`C = lqgtrack(kest,k,'1dof')`

forms a one-degree-of-freedom
LQG servo controller `C`

that takes the tracking
error *e* = *r* – *y* as
input instead of [*r* ; *y*], as
shown in the following figure.

The one-degree-of-freedom LQG servo controller state-space equations are

$$\begin{array}{l}\left[\begin{array}{c}\dot{\widehat{x}}\\ {\dot{x}}_{i}\end{array}\right]=\left[\begin{array}{cc}A-B{K}_{x}-LC+LD{K}_{x}& -B{K}_{i}+LD{K}_{i}\\ 0& 0\end{array}\right]\left[\begin{array}{c}\widehat{x}\\ {x}_{i}\end{array}\right]+\left[\begin{array}{c}-L\\ I\end{array}\right]e\\ u=\left[\begin{array}{cc}-{K}_{x}& -{K}_{i}\end{array}\right]\left[\begin{array}{c}\widehat{x}\\ {x}_{i}\end{array}\right]\end{array}$$

`C = lqgtrack(kest,k,...CONTROLS)`

forms
an LQG servo controller `C`

when the Kalman estimator `kest`

has
access to additional known (deterministic) commands *U _{d}* of
the plant. In the index vector

`CONTROLS`

, specify
which inputs of `kest`

are the control channels [

*U*;_{d}*r*;*y*] in the two-degree-of-freedom case[

*U*;_{d}*e*] in the one-degree-of-freedom case

The corresponding compensator structure for the two-degree-of-freedom cases appears in the following figure.

See the example Design an LQG Servo Controller.

You can use `lqgtrack`

for both continuous-
and discrete-time systems.

In discrete-time systems, integrators are based on forward Euler
(see `lqi`

for details). The
state estimate $$\widehat{x}$$ is either *x*[*n*|*n*]
or *x*[*n*|*n*–1],
depending on the type of estimator (see `kalman`

for
details).

For a discrete-time plant with equations:

$$\begin{array}{c}x\left[n+1\right]=Ax\left[n\right]+Bu\left[n\right]+Gw\left[n\right]\\ y\left[n\right]=Cx\left[n\right]+Du\left[n\right]+Hw\left[n\right]+v\left[n\right]\text{\hspace{1em}}\left\{\text{Measurements}\right\}\end{array}$$

connecting the "current" Kalman estimator to the LQR gain is optimal only when $$E\left(w\left[n\right]v{\left[n\right]}^{\prime}\right)=0$$ and *y*[*n*] does not depend on
*w*[*n*] (*H* = 0). If these conditions are not satisfied, compute the optimal LQG
controller using `lqg`

.