# AsymptoticOutputTracker

AsymptoticOutputTracker[sys,{f1,},{p1,}]

gives the state feedback control law that causes the outputs of the affine system sys to track the reference signals fi with decay rates pj.

AsymptoticOutputTracker[{sys,{out1,},{in1,}},]

specifies outputs outi and control inputs inj to use.

# Details and Options

• The system sys can be a StateSpaceModel or AffineStateSpaceModel.
• The reference signals fi should be pure functions of a single variable.
• The decay rates pi correspond to pole locations for the closed-loop system, and the number of poles pi is given by .
• The outputs {out1,} and inputs {in1,} are part specifications and by default are taken to be All.
• The number of outputs and reference signals must be the same.
• AsymptoticOutputTracker is based on FeedbackLinearize, and any residual dynamics must be stable for valid results. »

# Examples

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## Basic Examples(1)

A linear single-output system tracking a sinusoid:

The feedback law:

The closed-loop system:

After initial transients, the output tracks the reference:

## Scope(3)

An affine system tracking a piecewise constant signal:

The feedback law:

Simulate the closed-loop system:

Generate a trajectory passing through random points:

Design a feedback law that causes an affine system to track the trajectory:

Simulate the closed-loop system:

If there are more inputs than outputs, some of the feedback inputs are zero:

Specify the second input to use for control feedback:

Compute the closed-loop systems in each case:

Both the controllers achieve tracking:

## Applications(4)

Design a controller for a flexible joint to track a specified trajectory while carrying a load at one end: »

A model of the system:

The joint needs to be rotated from 5° to in 10 seconds:

The trajectory starts and ends with zero velocity and is smooth:

Obtain a computed torque control law:

The closed-loop system:

The joint follows the trajectory after initial transients and remains at after 10 seconds:

Compute a control law for a stepper motor to position a load at 1° in 0.1 seconds along the trajectory of a first-order system. A model of the motor: »

The reference trajectory:

The feedback law:

The closed-loop system has the desired response:

The glycolytic-glycogenolytic pathway, where the rates of metabolites , , and are taken as the manipulated variables, and , , , and are kept constant. Design a feedback law that maintains the values of the metabolites , , and at 0.2, 0.5, and 0.4: »

A model of the system:

A feedback law that maintains the values of , , and at 0.2, 0.5, and 0.4:

Simulate the closed-loop system:

The control effort:

Using Norrbin's model for the steering dynamics of a ship, design a control law that steers it with constant yaw rate and compare the control efforts for different ship speeds and lengths: »

A control law that steers the ship from 0° to 20° in 30 seconds and constant yaw rate:

The closed-loop system:

The simulation with specific values of ship speed and hull length :

The rudder deflection for various hull lengths:

The rudder deflection for various ship speeds:

## Properties & Relations(2)

The number of decay rates to be specified is determined by the sum of the vector relative orders:

This indicates that a single decay rate needs to be specified:

The decay rates for each output are assigned based on its relative order:

There are three decay rates needed for output 1 and two decay rates for output 2:

Specify slow rates for output 1 and fast rates for output 2:

Output 2 will track faster than output 1:

## Possible Issues(1)

For the feedback law to be effective, any residual dynamics must be stable:

The feedback law is not effective:

This is because the residual dynamics are unstable:

Wolfram Research (2014), AsymptoticOutputTracker, Wolfram Language function, https://reference.wolfram.com/language/ref/AsymptoticOutputTracker.html.

#### Text

Wolfram Research (2014), AsymptoticOutputTracker, Wolfram Language function, https://reference.wolfram.com/language/ref/AsymptoticOutputTracker.html.

#### CMS

Wolfram Language. 2014. "AsymptoticOutputTracker." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AsymptoticOutputTracker.html.

#### APA

Wolfram Language. (2014). AsymptoticOutputTracker. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AsymptoticOutputTracker.html

#### BibTeX

@misc{reference.wolfram_2023_asymptoticoutputtracker, author="Wolfram Research", title="{AsymptoticOutputTracker}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/AsymptoticOutputTracker.html}", note=[Accessed: 01-March-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_asymptoticoutputtracker, organization={Wolfram Research}, title={AsymptoticOutputTracker}, year={2014}, url={https://reference.wolfram.com/language/ref/AsymptoticOutputTracker.html}, note=[Accessed: 01-March-2024 ]}