A system with one feedback input and one measurement:

The feedback input and measurement are inputs to the estimator-regulator:

A system with one noisy input as well:

Noisy inputs are not inputs to the estimator-regulator:

A system with one feedback and one exogenous input:

The exogenous input is an input to the estimator-regulator:

A system where only some of the outputs are measured:

Only the feedback input and measured output are fed to the estimator-regulator:

Compute the gains using pole placement:

Assemble the estimator-regulator:

Compute the gains optimally:

Assemble the estimator-regulator:

Compute only the feedback gains optimally:

Compute the estimator gains using pole placement:

Assemble the estimator-regulator:

The feedback data object can be also specified:

A set of estimator gains:

Assemble the estimator-regulator:

A continuous-time StateSpaceModel:

The controller for a set of gains:

The closed-loop system:

Compute it automatically:

The poles of the closed-loop system:

A discrete-time StateSpaceModel:

The controller for a set of gains:

Its closed-loop system:

And poles:

A descriptor StateSpaceModel:

The controller has the same descriptor matrix:

Its closed-loop system:

And poles:

An AffineStateSpaceModel with a set of gains:

Its controller:

The closed-loop system:

Its poles are that of the Taylor linearized model:

A NonlinearStateSpaceModel with a set or gains:

Its controller:

The closed-loop system:

Its poles are that of the Taylor linearized system:

By default, EstimatorRegulator returns the controller comprising the estimator and regulator:

The controller can also be obtained as a property:

The estimator model:

The feedback gains model:

The closed-loop system:

The closed-loop poles:

The estimator-regulator feedback model:

In this model, the feedback input is fed back directly:

Assemble the estimator and regulator with feedback to get the same result as before:

The closed-loop system differs from the computed one only in the input matrix:

The design method:

Properties related to the input model and gains:

Get the controller data object:

The list of available properties:

The value of a specific property:

Design a tracking controller:

First design a state feedback controller and estimator:

Assemble the tracking controller:

The closed-loop system tracks the reference signal :

Design a tracking controller for a discrete-time system:

First design a state feedback controller and estimator:

Assemble the tracking controller:

The closed-loop system tracks the reference signal :

Track multiple outputs:

Design an optimal state feedback controller:

And an estimator using pole placement:

Assemble the tracking controller:

The closed-loop system tracks two different reference signals:

Compute the controller effort:

Design an optimal state feedback controller:

And an estimator:

Assemble the tracking controller:

The controller model:

The controller inputs:

The controller effort:

Track a desired reference signal:

The reference signal is of order 2:

And sinusoidal:

Set the reference as as sum of sinusoids:

Design a controller to track one output of a first order system :

The dimensions of the state weighting matrix qq is k+m q: :

Compute the controller:

The closed-loop system tracks the reference:

Compute the closed-loop system based on a linear design:

The closed-loop system based on the nonlinear model:

Compare the response of the two designs:

Assemble the merged closed-loop system of a plant with an EstimatorRegulator:

The unmerged closed-loop system:

When merged, it gives the same result as before:

Merge the closed-loop system explicitly:

Create a closed-loop system with a desired name:

The closed-loop system has the specified name:

The name can be directly used to specify the closed-loop model in other functions:

The simulation result:

Design a controller for a partially controllable double cart-spring system:

A model of the system:

The system is not controllable:

The controllable subsystem:

The response of the open-loop system:

Compute a set of optimal feedback gains:

And a set of estimator gains:

Assemble the estimator-regulator:

The closed-loop system:

The horizontal positions of the masses are controlled:

While the vertical position of is not:

The controller model:

The controller effort:

Dampen the oscillations of a three mass system:

A model of the system:

The open-loop response is highly oscillatory:

Compute a set of optimal regulator gains:

And a set of estimator gains:

Assemble the estimator-regulator:

The closed-loop system:

The oscillations are suppressed by the feedback controller:

The controller model:

The controller effort:

Dampen the oscillations of a coupled pendulum:

A model of the system:

The undamped open-loop response:

Compute a set of estimator gains:

Compute a set of optimal gains:

Assemble the estimator-regulator:

The closed-loop system:

The pendulums are damped:

The controller model:

The control effort:

source: https://fcichos.github.io/CompSoft21/notebooks/L6/2_coupled_pendula.html

Stabilize the orbit of a spacecraft near a circular orbit of mean motion ω:

A model of the system:

Discretize the system:

The open-loop response is unstable:

Compute a set of regulator gains:

And a set of estimator gains:

Assemble the estimator regulator:

The closed-loop system:

The satellite's orbit is stabilized:

The controller model:

The control effort:

Stabilize a helicopter:

A model of the system:

The system is unstable:

Compute a set of regulator gains:

And a set of estimator gains:

Assemble the estimator-regulator:

The closed-loop system:

The helicopter is stabilized:

The controller model:

The control effort:

Regulate the blood-glucose level in the human body:

The Bergman's model of the system:

The system is slow with poles close to the y-axis:

The glucose level in the open-loop system takes a long time to settle down:

Compute a set of optimal gains:

And a set of estimator gains:

Assemble an estimator-regulator:

The closed-loop system:

The blood-glucose level settles down much faster:

The controller model:

The control effort:

Regulate the biomass concentration in an isothermal constant-volume fermenter: »

A model of the fermenter:

The biomass concentration of the unregulated system:

Compute a set of estimator gains:

Compute a set of regulator gains:

The estimator regulator:

The closed-loop system:

The biomass concentration is regulated:

The control effort:

Improve the response of a mixing tank process:

A model of the system:

The output response takes to stabilize:

The system’s fast and slow modes are controllable:

Compute a set of state feedback gains:

And a set of estimator gains:

The estimator regulator:

The closed-loop system:

The response of the closed-loop system only takes to stabilize:

The controller model:

The control effort:

Regulate a DC-motor driven by a buck converter:

A model of the system:

The system has poles close to the imaginary axis:

Therefore the response of the system to an initial perturbation is slow:

Compute a set of regulator gains:

And a set of estimator gains:

Assemble the estimator regulator:

The closed-loop system:

The closed-loop system with the estimator regulator in the feedback path:

The controller model:

Assess the control effort:

Dampen the response of the terminal voltage of a power system:

A model of the system:

The system has two lightly damped poles:

The open-loop response is therefore highly oscillatory:

Compute a set of regulator gains:

And a set of estimator gains:

Assemble the estimator-regulator:

Obtain the closed-loop system:

The closed-loop response to a perturbation in the synchronous machine is damped and is faster:

Compute the controller model

The control effort:

Control a quadcopter drone’s altitude:

A model of the system:

Without any input, the drone is in free fall:

A set of optimal regulator weights:

Compute the optimal regulator:

And a set of estimator gains:

Assemble the estimator-regulator:

The closed-loop system:

The drone's altitude is regulated:

The controller model:

The controller inputs:

The controller effort: