As an example of LQG design, consider the following regulation problem.

The goal is to regulate the plant output *y* around zero. The input
disturbance *d* is low frequency with power spectral density (PSD)
concentrated below 10 rad/s. For LQG design purposes, it is modeled as white noise driving a
lowpass filter with a cutoff at 10 rad/s, shown in the following figure.

For simplicity, this noise is modeled as Gaussian white noise with variance of 1.

The following figure shows the Bode magnitude of the shaping filter.

**Bode Magnitude of the Lowpass Filter**

There is some measurement noise *n*, with noise intensity given by

$$E({n}^{2})=0.01$$

Use the cost function

$$J(u)={\displaystyle {\int}_{0}^{\infty}(10{y}^{2}+{u}^{2})dt}$$

to specify the tradeoff between regulation performance and cost of control. The following equations represent an open-loop state-space model:

$$\begin{array}{l}\dot{x}=Ax+Bu+Bd\text{\hspace{1em}}(state\text{\hspace{0.17em}}equations)\\ y=Cx+n\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}(measurements)\end{array}$$

where (*A*,*B*,*C*) is a state-space
realization of $$100/({s}^{2}+s+100)$$.

The following commands design the optimal LQG regulator *F*(*s*) for this problem:

sys = ss(tf(100,[1 1 100])) % State-space plant model % Design LQ-optimal gain K K = lqry(sys,10,1) % u = -Kx minimizes J(u) % Separate control input u and disturbance input d P = sys(:,[1 1]); % input [u;d], output y % Design Kalman state estimator Kest. Kest = kalman(P,1,0.01) % Form LQG regulator = LQ gain + Kalman filter. F = lqgreg(Kest,K)

These commands returns a state-space model `F`

of the LQG regulator
*F*(*s*). The `lqry`

,
`kalman`

, and `lqgreg`

functions perform discrete-time LQG
design when you apply them to discrete plants.

To validate the design, close the loop with `feedback`

, create and add
the lowpass filter in series with the closed-loop system, and compare the open- and
closed-loop impulse responses by using the `impulse`

function.

% Close loop clsys = feedback(sys,F,+1) % Note positive feedback. % Create the lowpass filter and add it in series with clsys. s = tf('s'); lpf= 10/(s+10) ; clsys_fin = lpf*clsys; % Open- vs. closed-loop impulse responses impulse(sys,'r--',clsys_fin,'b-')

These commands produce the following figure, which compares the open- and closed-loop impulse responses for this example.

**Comparison of Open- and Closed-Loop Impulse Response**