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.


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:


open allclose all

Basic Examples  (1)

Construct an LQG regulator for a system with one measured output and one feedback input:

Scope  (4)

An LQG regulator for a system with one measurement and two feedback inputs:

An LQG regulator for a system with feedback, deterministic input, and stochastic inputs:

Design an LQG regulator for a discrete-time system:

Find the optimal regulator for a descriptor state-space model:

Applications  (2)

An LQG regulator for a SISO system:

The closed-loop 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:

Design an LQG regulator:

Find the closed-loop system:

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:

Introduced in 2010
Updated in 2012