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.


specifies outputs outi and control inputs inj to use.

Details and Options


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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:

Introduced in 2014