Documentation |
Form linear-quadratic-Gaussian (LQG) regulator
rlqg = lqgreg(kest,k)
rlqg = lqgreg(kest,k,controls)
lqgreg forms the linear-quadratic-Gaussian (LQG) regulator by connecting the Kalman estimator designed with kalman and the optimal state-feedback gain designed with lqr, dlqr, or lqry. The LQG regulator minimizes some quadratic cost function that trades off regulation performance and control effort. This regulator is dynamic and relies on noisy output measurements to generate the regulating commands.
In continuous time, the LQG regulator generates the commands
$$u=-K\widehat{x}$$
where $$\widehat{x}$$ is the Kalman state estimate. The regulator state-space equations are
$$\begin{array}{l}\dot{\widehat{x}}=[A-LC-(B-LD)K]\widehat{x}+Ly\\ u=-K\widehat{x}\end{array}$$
where y is the vector of plant output measurements (see kalman for background and notation). The following diagram shows this dynamic regulator in relation to the plant.
In discrete time, you can form the LQG regulator using either the delayed state estimate $$\widehat{x}[n|n-1]$$ of x[n], based on measurements up to y[n–1], or the current state estimate $$\widehat{x}[n|n]$$, based on all available measurements including y[n]. While the regulator
$$u\left[n\right]=-K\widehat{x}\left[n|n-1\right]$$
is always well-defined, the current regulator
$$u\left[n\right]=-K\widehat{x}\left[n|n\right]$$
is causal only when I-KMD is invertible (see kalman for the notation). In addition, practical implementations of the current regulator should allow for the processing time required to compute u[n] after the measurements y[n] become available (this amounts to a time delay in the feedback loop).