lqgreg (Control System Toolbox)

Form LQG regulator given state-feedback gain and Kalman estimator

Syntax

rlqg = lqgreg(kest,k)
rlqg = lqgreg(kest,k,'current')    % discrete-time only

rlqg = lqgreg(kest,k,controls)

Description

lqgreg

forms the 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 (see page 7-10 for details).

In continuous time, the LQG regulator generates the commands

where

is the Kalman state estimate. The regulator state-space equations are

where

is the vector of plant output measurements (see kalman for background and notation). The diagram below shows this dynamic regulator in relation to the plant.

In discrete time, you can form the LQG regulator using either the prediction

based on measurements up to

, or the current state estimate

based on all available measurements including

. While the regulator

is always well-defined, the current regulator

is causal only when

is invertible (see kalman for the notation). In addition, practical implementations of the current regulator should allow for the processing time required to compute

once the measurements

become available (this amounts to a time delay in the feedback loop).

Usage

rlqg = lqgreg(kest,k) returns the LQG regulator rlqg (a state-space model) given the Kalman estimator kest and the state-feedback gain matrix k. The same function handles both continuous- and discrete-time cases. Use consistent tools to design kest and k:

Continuous regulator for continuous plant: use lqr or lqry and kalman.
Discrete regulator for discrete plant: use dlqr or lqry and kalman.
Discrete regulator for continuous plant: use lqrd and kalmd.

In discrete time, lqgreg produces the regulator

by default (see "Description"). To form the "current" LQG regulator

instead, use the syntax

rlqg = lqgreg(kest,k,'current')

This syntax is meaningful only for discrete-time problems.

rlqg = lqgreg(kest,k,controls) handles estimators that have access to additional known plant inputs

. The index vector controls then specifies which estimator inputs are the controls

, and the resulting LQG regulator rlqg has

and

as inputs (see figure below).

Note:
Always use positive feedback to connect the LQG regulator to the plant.

Example

See the examples "Control Design Tools" in the manual and "LQG Regulation" in the manual.