LQGRegulator
LQGRegulator[{ssm,sensors,finputs},{w,v,h},{q,r,p}]
constructs the optimal feedback regulator for the StateSpaceModel ssm using noisy measurements sensors and feedback inputs finputs. The process, measurement, and cross-covariance matrices are w, v, and h; and the state, input, and state-input weighting matrices are q, r, and p.
LQGRegulator[{ssm,sensors,finputs,einputs},{…},{…}]
specifies einputs as the exogenous deterministic inputs.
Details and Options
- The standard state-space model ssm can be given as StateSpaceModel[{a,b,c,d}], where a, b, c, and d represent the state, input, output, and transmission matrices in either a continuous-time or a discrete-time system:
-
continuous-time system 
discrete-time system - The descriptor state-space model ssm can be given as StateSpaceModel[{a,b,c,d,e}] in either continuous time or discrete time:
-
continuous-time system 
discrete-time system - LQGRegulator also accepts nonlinear systems specified by AffineStateSpaceModel and NonlinearStateSpaceModel.
- For nonlinear systems, the operating values of state and input variables are taken into consideration when constructing the LQGRegulator.
- The input
can include stochastic inputs 
, feedback inputs 
, and exogenous deterministic inputs 
. - The arguments finputs and einputs are lists of integers specifying the positions of
and 
in 
. - The output
consists of the noisy measurements 
as well as other outputs. - The argument sensors is a list of integers specifying the positions of
in 
. - LQGRegulator[{ssm,sensors,finputs},{…},{…}] is equivalent to LQGRegulator[{ssm,sensors,finputs,None},{…}].
- If not specified, h and p are assumed to be zero matrices.
- Block diagram of the continuous-time system with its regulator:
- Block diagram of the discrete-time system with its regulator:
- The system with the regulator has the following block diagram:
Examples
open allclose allBasic Examples (1)
Scope (4)
Applications (2)
An LQG regulator for a SISO system:
The closed-loop output response:
A descriptor system with quadratic noise and cost matrices:
Create Gaussian noise sequences:
Interpolate the sequences to get noise signals:
The open-loop response is slow to settle:
The closed-loop system has a faster settling time:
Properties & Relations (2)
The closed-loop poles are the poles of the state-feedback and estimator subsystems:
The poles of the closed-loop system with optimal state feedback:
The poles of the optimal estimator:
The pole locations of the closed-loop system and the feedback-estimator system coincide:
Construct an LQG regulator from the optimal state-feedback gains and the Kalman estimator:
The LQ estimator gains and the Kalman estimator model:
Create the estimator-regulator manually:
LQGRegulator gives the same result: